Make a breading assembly line with 3 separate dishes: flour seasoned with salt and pepper, milk and eggs whisked together and panko seasoned with salt and pepper. Season the chicken breasts with salt and pepper, then bread them by dredging in the flour, then dipping in the egg mixture and finally coating in the crumbs. Set aside.
Melt 4 tablespoons of the butter with 4 tablespoons of the olive oil in a large skillet over medium heat. Brown the chicken breasts on the first side for 3 to 4 minutes, being careful not to burn them. Flip and brown them for another 3 to 4 minutes. Remove to a rack placed over a baking sheet and transfer to the oven to finish cooking, about 5 minutes.
Remove the chicken from the oven and turn on the broiler. Top the chicken with the mozzarella slices and broil until the cheese is melted.
Heat a small skillet over medium-high heat and add the remaining 1 tablespoon butter and 1 tablespoon olive oil. Add the tomatoes, sprinkle with salt and pepper and toss to cook quickly until the tomatoes start to break apart, about 2 minutes. Add the spinach and garlic, then stir/toss to wilt the spinach, 2 to 3 minutes. Spoon the tomato/spinach mixture over the chicken and serve.