Beat together the eggs, milk and Parmigiano-Reggiano in a medium bowl until combined. Set aside.

Divide the mozzarella evenly between 2 slices of the white bread. Put the remaining slices of bread on top of the mozzarella.

In a large skilet, add enough canola oil to be 1/2" deep. Heat over medium-high heat until the oil shimmers. A good way to test this if you can't tell by sight is by dropping a bit of the egg, milk, and cheese mixture into the oil. If it starts bubbling and frying up, you are ready!

Dredge each sandwich in flour and shake off the excess. Then dip each sandwich in the egg and cheese mixture. Work in batches, frying each sandwich individually. Secure each sandwich with toothpicks before adding to the pan. Once golden brown, remove the toothpicks, flip and insert the toothpicks in the other side. When the both sides of the sandwich are golden brown, remove from oil to paper towel lined plate to drain. Repeat this process with the other sandwich. Cut the sandwiches into quarters, arrange on a serving platter and serve with the sauce.

To prepare the sauce, combine the olive oil, garlic, and crushed red pepper in a small saucepan over medium heat until the garlic is golden brown. Add the lemon juice and stir to combine. Stir in the butter and season with salt, to taste. Once the butter is melted, remove from heat and serve immediately.