PREPARATION: Cut the ends off one of the lemons and pare away the rind and all the white pith down to the flesh. Cut lemon into paper-thin rounds, removing any seeds. Whisk the eggs. Whisk in the Parmesan cheese, 3 tbs flour, and enough milk to make a batter the consistency of heavy cream. Cut the fish across the grain into 1/4-inch thick medallions. Sprinkle with salt and pepper. Recipe can be completed to this point several hours ahead.
Panfry them in the butter and oil, turning them once, until the coating is golden brown and the fish just tests done, about 3 minutes total. Transfer the fish to warm plates. Make a lemon butter sauce by squeezing the juice from the second lemon into the frying pan and stirring with a wooden spoon to deglaze the pan. Add salt and pepper to taste. Top each piece of fish with a few capers and lemon