Season both sides of the chicken breasts with salt and pepper.
Prepare three shallow dishes: place flour on one plate, whisk the egg with 1 tablespoon water in a bowl, and put bread crumbs on a second plate.
Dredge each chicken piece in flour, shake off excess, dip into the egg mixture, then coat evenly with bread crumbs.
Heat the butter and olive oil in a large skillet over medium-high heat until the butter is melted and sizzling.
Add the breaded chicken and cook 3 to 4 minutes per side, until golden brown and cooked through; transfer to a plate and set aside.
Add the minced garlic to the pan drippings and cook about 1 minute until fragrant.
Reduce the heat slightly, pour in the chicken broth and heavy cream, and simmer 2 to 3 minutes.
Stir in the lemon juice and the cornstarch slurry, simmer about 1 minute until the sauce slightly thickens.
Stir in the Parmesan cheese and capers until combined.
Return the chicken to the skillet to warm in the sauce for a minute, then remove from heat and garnish with lemon slices and chopped parsley before serving.