Season chicken on both sides with salt and pepper.
Mix the flour, salt and pepper in a plate.
In a second plate (or bowl), beat the egg and 1/2 tablespoon of water together.
On a third plate, add the breadcrumbs.
Dip each chicken breast first in the flour, then dip in the egg then the bread crumbs.
Cook the chicken
Heat 1 tablespoon of olive oil in a large saute pan over medium to medium-low heat.
Add the chicken breasts and cook for 2 minutes on each side, until browned. Put them in the preheated oven, and bake for 5 to 10 minutes while you make the sauce.
Make the piccata sauce
Dry the skillet with a dry paper towel.
Over medium heat, melt 1 tablespoon of the butter and then add the lemon juice, wine, stock the reserved lemon halves, 1/2 teaspoon salt, and 1/4 teaspoon pepper.
Boil over high heat until reduced in half, about 2-3 minutes.
Serve
Remove from the heat, add the rest of the butter and stir gently. Add the capers.
Discard the lemon halves and serve a piece of chicken on each plate, with the sauce on the side, or poured sparingly over the chicken.
Notes
Bring the extra sauce to the table, because it's good!