Preheat the oven to 180°C (350°F/Gas 4). Wipe the chicken breasts clean with paper towels. Cut a long deep pocket in each one through the thickest side, being careful not to cut all the way through or the filling will ooze out. Scatter a little salt and pepper over both sides of the chicken. Mix 1 tablespoon of the parsley with the thyme, garlic, anchovies, capers and mozzarella in a small bowl. Divide among the chicken breasts and stuff well into the pockets. Use 2 toothpicks to seal off each opening.
Pulse the bread in a food processor to give uniform soft crumbs. Remove to a plate and mix in the remaining 2 tablespoons of parsley and the parmesan. Lightly beat the egg in a wide bowl with some salt and black pepper. Dip the chicken breasts in the egg and then pat gently into the crumbs, helping them stick with your hands.
Put 2 tablespoons of the oil and the butter in a baking dish just large enough to hold the chicken snugly. Add the chicken, drizzle over the remaining 2 tablespoons of oil and bake for 30 minutes or longer, until deep golden and slightly crispy. Remove from the oven and rest, covered with foil, for 5 minutes or so. Remove the toothpicks and slice each breast on the diagonal into about 6 slices. Put on serving plates, drizzle each with a little juice from the pan and give a final grind of black pepper. Serve with a squeeze of lemon juice if you like.