This question was asked on metafilter, but we lacked the expertise to give it the appropriate historical context. Indeed, according to Gaston's poetic testimony*,
"When I was a lad
I ate four dozen eggs
Ev'ry morning to help me get large
And now that I'm grown
I eat five dozen eggs
So I'm roughly the size of a barge!"
With Gaston consuming (60 eggs * 7 days = 420 eggs / week), and a modern laying hen producing around 4 eggs / week in a backyard environment, we would require a flock of just over a hundred modern hens in prime condition to support adult Gaston's nutritional habits. However, while I imagine that 16th century French laying hens would have been less productive lacking modern breeding, what exactly were their laying rates and how many more would be required? Gaston appears to be a prominent and prosperous citizen in his village, but how significant would the economic activity required to maintain his breakfast habits be? Also, were the eggs produced by 16th century French hens significantly smaller than modern eggs, thus reducing the Cool-Hand-Luke-esque nature of Gaston's daily challenge, but if so how much smaller?
Similarly, modern chickens will only lay eggs when exposed to the sufficient amounts of light present in the spring and summer months, would there have been feasible ways for 16th century French villagers to convince the chickens they had access to to lay eggs during the late fall/early winter months that Gaston is singing in?
Goddamn. This has to be the most random-ass hilarious question I might be somewhat capable of answering.
As per usual, disclaimer: I'm not an expert on poultry science. Culinary history is a hobby field for me, and I've occasionally looked into the surge of backyard chicken-keeping in the U.S. and Canada because apparently I have nothing better to do. Heritage breeds are very popular for small-time poultry keepers, and people are generally quite interested in the history behind these breeds. This may not be more than a decent guess, and even that might be kind.
You're absolutely right that modern chickens have been bred to be more productive than their ancient forebears. The modern laying chicken according to PSU's College of Agricultural Sciences is a White Leghorn for white eggs or a sex-linked hybrid of New Hampshire Red/Barred Plymouth Rock descent for brown eggs. (And by sex-linked hybrid, we mean a chick that can be sexed the day of its birth; the unfortunate reality is that cockerel chicks are of little value to the industry and are typically killed.) These birds exist solely to produce eggs, whether for consumption or breeding, and a hen in the prime of her life (20 to 85 weeks) can crank out 300+ per year.
300 eggs would have been an insane number for a bird in ... uh, whenever Beauty and the Beast is set. We'll get to that in a moment. Suffice it to say that the average small farmer in times past generally couldn't afford, or didn't have the space, to breed both meat and laying chickens. They were also reluctant to slaughter a bird that was still productive even if that productivity was declining, which is why the majority of chicken recipes from before the 20th century assume the use of an older bird. It is not a mistake that just about every poultry-keeping culture you can name had an abundance of recipes for old birds (e.g., the incredible array of chicken soups from around the world) and relatively few for younger ones.
In terms of actually answering your question, I think the logical progression is:
So when is Beauty and the Beast actually set? I googled this because I wasn't sure and just wanted a frigging date. Disney, you're killing me. The original story comes from a French fairy tale either written or transcribed by Gabrielle-Susanne Barbot de Villeneuve in 1740. However, if we assume the matches that Lumiere uses in the animated 1991 version date the film, then it can't have happened before 1805 when the match was invented (conveniently enough, in Paris). But then there's the Eiffel Tower that the magic forks assemble during "Be Our Guest," and construction on the Eiffel Tower was between 1887 and 1889. I haven't seen the live-action version yet and I don't know the degree to which it differs.
Luke Evans (who plays Gaston in the live-action adaptation) said his characterization of Gaston assumes he was around 16 in 1740. Evans is 37 now, so let's argue that Gaston is somewhere between his late 20s and late 30s in the film, which dates it to 1750-1763. The fashion design seems to put the film squarely in 18th-century France. Other materials distributed by Disney vaguely reference "Bourbon France," which means sometime between 1589 (Henry IV) and 1792 (French Revolution and boom goes the monarchy) and I assume not the brief-lived Bourbon restoration (1814 or 1815 depending on how you choose to define "restored" until 1830).
Mid-18th century it is. My head hurts and we are moving the fuck on.
Where is Beauty and the Beast set? For the 1991 film, Disney's art department used both the Loire Valley (central France) and the towns of Riquewihr and Ribeauvillé as inspirations for the setting. This is a bit problematic as both towns are several hundred miles to the east of the Loire Valley, but whatever. Unfortunately, I'm not sure if anyone on the live-action film has commented on the specific setting. We'll say somewhere in north-northeastern rural France where: a). people are forced into choirs at birth, and: b). can harmonize in adulthood at will.
What breeds of chicken were common in that era and place? This is very tough. France has produced an incredible variety of livestock, and unfortunately much of the historical record concerning it has yet to be translated. We'll do the best we can with the information available in English. There are a few we can eliminate right off the bat (e.g., the Estaires didn't appear in Europe until the late 19th century, the Coucou de Rennes is from the wrong region, etc.), but quite a few remain. However, there are at least three ancient French breeds that comfortably predate the mid-18th century and would have been common in northern/northeastern France:
There are definitely more, but these are the only ones I can absolutely confirm after running through my library and cross-checking online.
Do we have records of these breeds' productivity? If we do, I'm not sure that records of their mid-18th century production exist in English, but let's take the figure of 3 eggs per week between March and October. Gaston's already in trouble. The modern backyard/hobby farmer you cite who's getting 4 eggs/week per bird is nearly always raising chickens as a sort of combination pet and egg producer. The growth of backyard chicken-keeping has prejudiced the market in favor of breeds that fit this profile, e.g., the Buff Orpington, the Rhode Island Red, and Ameraucana. For that matter, even 4 eggs/week is a bit generous for these birds; not a lot of people have the stomach to slaughter their chickens as their productivity declines, so 4 eggs/week assumes a young bird that's eating well and is at her most productive. There is a French breed, the Marans, that can match this output, but it's not in contention for feeding Gaston here as it was developed in the 20th century.
Again, without hard records on exactly how productive the La Flèche, Houdan, and Bresse Gauloise were during the mid-18th century, I can't say with 100% certainty that 3 eggs/week is what you'd get. I will say based on the scanty information I've seen on hen productivity before the industrialization of agriculture is that it's a very plausible number.
All right. Gaston is probably -- or at least plausibly -- stuck with birds that were laying 3 eggs/week. If that's the case and he's eating 420 eggs per week, then he will need a flock of 140 chickens -- or realistically a bit more to account for randomness and the occasional loss of a bird to predators or theft -- to satisfy his protein intake. Additionally, this also assumes that the birds in question are eating well, and that will depend on the farmer's income (can he afford to supplement their foraging?) and their environment (are they well-supplied with insects and plants, or are there too many birds for the farmer's land to support?).
Also, he's going to have a hard time between October and March, because all of these breeds would have been subject to the normal seasonal cycle, and egg output would have declined or stopped entirely. Due to this, he would have needed substantially more than 140 chickens so he could pool eggs ahead of time to get himself through the winter (and eggs, if not stripped of their protective membrane, will last for a few months).
SOURCES:
TL:DR: 140+ chickens. I'm off to go make an omelet.
Here is your answer:
This 1849 agricultural journal which delves about exporting agricultural goods to the UK mentions that an economist has calculated that a hen produces 120 eggs a year.
Since there has been no major changes in agricultural practice in France between the mid 18th (Beauty and the Beast is in the 18th century, not the 16th) and the early 19th, we can be confident that egg production was similar then. We can lower the figure a bit, to 104 eggs a year, or two eggs a week.
Hence Gaston would need 210 hens to produce his eating habits. In the same page quoted, it says "chaque famille possede une douzaine de poules" (each family possesses a dozen of hens) which gives you a scale. This means that a hundred family village would require extra 2 hens per family to feed Gaston for breakfast.
Follow-up question: were there even barges in 16th century France?
We know from the opening song that Gaston's eating habits have wreaked havoc in the local egg economy (eggonomy?):
I need six eggs; that's too expensive!
And we actually had a clue to this even earlier in the opening song, thanks to these lines about the baker:
There goes the baker with his tray, like always The same old bread and rolls to sell
Follow up question: Is it possible that Gaston has such an impact on this towns economy that it is even impacting the ability to innovate by local businesses, as the baker might not be able to afford eggs for his business?
What about duck eggs? It never specifically says chicken eggs could be duck eggs or another type of egg?
How much would an egg cost during this time period if one were to be purchasing rather than raising?
I'm assuming to buy that many eggs though one would have to be wealthy & there's nothing to suggest that Gaston has considerable wealth.