I understand Homer may not have been a real individual in the same vein of Shakespeare; supposing he was is there any document or oft told story to back this up?
Nobody is even sure about when Homer actually lived, with sources varying between 1,000 to 800 BC.
Now, butter had been invented by this point but it's very different from its modern counterpart due to the lack of refrigeration. I would say that someone of that period is much more likely to be eating clarified, liquid butter similar to Ghee. And it would have been sheep or goat's milk that would have been used to make the butter, not cow milk.
Mizithra meanwhile is a fresh goat's cheese, and it would have been available in the time period; perhaps even in a form that bears rather close resemblance to its modern counterpart. For reference, Pecorino Romano - another ewe cheese - was known to have been eaten by the Roman Legionnaires and is still used to this day.
That said, I'm guessing the dish the restaurant claims Homer ate was probably something like this:
Which is almost certainly not what Homer would have eaten. Solid butter simply wasn't common in that period and "browning" it while melting it was a post-refrigeration technique. More importantly, pasta would not be mentioned in historical writing until the 1st Century AD - a thousand years after Homer - and even then this was the fried and not boiled variety.
Hence, I suspect that if anyone ever did eat a meal of butter and Mizithra cheese, it would probably not be a pasta dish. More likely is that it's some bread with liquid butter as the oil, and Mizathra as the cheese - all in all something not dissimilar to the Roman Army ration of bread, olive oil, and Pecorino Romano cheese.