Preheat oven to 400°F (204°C). Place a sheet pan on the middle rack of the oven and let it heat up.
Take a large bowl and combine flour, bread crumbs, salt, and pepper in it. Stir them until everything is evenly mixed together. Make sure that the breadcrumbs are evenly distributed.
Take a medium-sized bowl and whisk 2 to 3 large eggs in it until smooth and creamy. Beat the eggs properly until they become completely homogenous. Place the flounder on the sheet pan and bake for 12 minutes (until lightly golden brown). Turn them over halfway through baking.
Heat butter and olive oil in a sauté pan on medium heat. Dip each side of the flounder in the egg mixture (using tongs) and then coat it with the flour-breadcrumbs mix. Keep doing this until all 6 fillets are coated properly with the flour-breadcrumb mix.
Place the flounder on the sauté pan and fry for 3 to 4 minutes (each side). Make sure that each side is golden brown before you flip them over. Remove the flounder from the sauté pan and place them on a plate lined with paper towels to absorb any excess oil.
Arrange the flounders on a plate and sprinkle some lemon juice over them for added flavor. Serve with arugula salad and Parmesan cheese shavings, along with a side of freshly made Lemon Vinaigrette.