Professor Resor of Eastern Kentucky University has a great article that discusses the topic of what an average European peasant's diet looked like (it even gives example diets, modern-day comparisons, and lots of juicy citations).

While you should definitely read the article, a quick summary would be that most of the carbohydrates were provided by things like wheat, rye, oats, and barley. Protein came from beans, peas, and lentils, with meat such as fish, poultry, beef, and pork being a rarer delicacy, usually consumed only on special occasions. Seasonal fruits and vegetables helped round out the diet. Ale was also a staple that helped provide the necessary calories (it's important to note its alcohol content was lower than that of more modern brews; lest you think medieval peasants were bombed out of their gourd all the time). As you inferred in your post title, food could become scarce in winter and early spring, and a lack of adequate food reserves could mean serious trouble to a peasant.

It's also important to note that the average peasant was a bit shorter, although he/she still required about 2000 calories per day, and that's before taking labor requirements into account (a peasant's work day could last upwards of 12 hours).

It's not hard to understand how the introduction of potatoes from America helped contribute to a European population boom. The humble tuber was responsible for around a quarter of the population boom in Europe between the 1700's to the 1900's. By the mid 1700's almost a third of Ireland's population was dependent upon the crop, and when it failed, people died.

I also recall a fascinating article by Malcolm Gladwell that explored the cultural impact that a society's staple crop could have upon it. While I'm probably doing a terrible job of summarizing, the major crux of his argument was that a European peasant was likely to grow about the same number of crops regardless of how much or how little effort they put into the job. On the other hand, the effort that Asian rice farmers exerted in their farming could have a massive impact on a crop's success. Bumper crops that weren't required for subsistence could be sold for profit. Additionally rice-farming is a task that benefits from knowledge, which would of course make education quite valuable. Therefore the average Asian rice farmer had much more to gain from hard work and education than the average European farmer.

EDIT: Final comment removed due to moderator request.

I also wanted to ask this question. Specifically, how different was Hungarian food without Paprika? What about Indian food without Curry? Thai and Vietnamese cuisine also use the chili pepper extensively.

Much of the cuisine we think of most is a relatively modern invention. Did the same food exist but without the hotness from the chili pepper? What cuisine where these places famous for before the Columbian Exchange?

There's a small section in Asa Briggs' Social History of England about the food eaten in the household of Simon de Montfort.

"While the poor were eating fairly simply, eating daily up to five pounds of coarse (but nutritious) bread made out of wholemeal barley or maslin (wheat and rye or beans) together with some meat, cheese, milk and butter, vegetables and lots of weak ale, the de Montfort household, which also consumed vast quantities of bread was supplementing it amply with meat, fish, and delicacies like pike and partridge.

Rice was so special that it was kept under lock and key. Spices were expenses ... but were considered necessary. Sugar could fluctuate in price from month to month and the weight of sugar consumed in a year was only just larger than the weight of pepper [honey seems to have been often used where we'd use sugar]. Many of the dishes on the table at this time were not only highly flavoured, but also highly coloured, either by bright yellow saffron or, perhaps less attractive, crimson blood. Gascon wine, which was imported in very large quantities from England's empire in France, was a major drink and the household brewed its own ale, though hops were not used in ale-making as early as the thirteenth century. Fruit and vegetables from the orchard and garden were not itemised in the accounts: they were taken for granted.

There are a few sites with collections of medieval recipes. I've made a good pease porage from one. I think this site is the best.

What you don't get from either of those sources, though, is the seasonality of of produce or the typical diet of different social groups.

A related question: could anyone comment on what Italian food was like before the introduction of the tomato? Also, how quickly did "New World foods" spread in Europe? At what point were potatoes as Irish as Guinness instead of a foreign food?

There's a lot of interesting discussions in here but I don't see anything answering OP yet. Would there have been other squashes and parsnips in Europe?

how did they make the beer?

According to Ferdinand Braudel most European peasants ate from a communal bowl of gruel with veg, tubers, and chestnut stews, with the occasional meat offcut thrown in. Which they would soak up with stale bread.

Coincedentally, I recently downloaded "Nutrition and the Early-Medieval Deit" by Kathy L. Pearson. This refers to an earlier period than the one in question (AD 400-900), but should still be generally applicable.

It depended a lot on which region you're looking at, but the staple vegetables included "root vegetables such as burdock, beets, carrots, turnips, and parsnips." Cabbage is very nutritious, as is kohlrabi. Lettuce, endive, arugula, cress, parsley, and lovage were also grown, as were members of the onion family like onion, chives, leeks, garlic, and shallots.

They grew cucumbers and celery, fennel, and radishes. They also grew gourds and squashes, although this source isn't clear about what kinds. There were a lot of fruits, including apples and pears.

By 1200, it's possible that other staples had been introduced from West Asia, but this is a basic list of vegetables in Europe at around AD 900, at least.

Question - do you mean vegetables specifically, or any type of Winter food in Western Europe?

Antidotes abound, but no one actually answering the question. I'm curious myself about this. Seems that so many of the staple veggies around the world are new world. What was around before then.