In a small bowl, combine basil, olives, cherry tomatoes, oil, salt, pepper, and lemon. Mix gently and refrigerate until chicken is ready to serve.
Heat 2 tablespoons oil in a large saucepan over medium-low heat. Add the onions and garlic and cook for about 8 minutes, stirring frequently. Add the plum tomatoes, tomato paste, sugar, salt, oregano, thyme, and pepper and mix thoroughly. Cook over medium-low heat until the sauce thickens and reduces in volume—18-20 minutes. Add in the butter and stir until it melts. Reduce heat to keep warm and set aside.
Place flour and eggs in separate, shallow dishes. Toss the breadcrumbs and Parmesan together in a shallow dish. Set aside.
Using a Ziplock bag, place chicken inside and tenderize with a rolling pin, until the breasts are all about the same thickness. Season chicken with salt and pepper.
Working one chicken breast at a time, dredge the chicken in flour, then dip in the eggs, and finally, coat it in the breadcrumbs. Repeat these steps for the remaining chicken.
Prepare an oiled baking dish for baking the chicken and set aside.
Heat about 3 tablespoons oil in a large skillet over medium-high heat. When oil is hot and shimmery, place two chicken breasts in and cook until breading is golden brown—3-4 minutes each side. Repeat for the remaining chicken breasts (adding additional oil as needed). Move chicken to a baking dish in a single layer, with 2 tablespoons sauce underneath each breast.
Top chicken with about 1⁄3 cup sauce, then top with a piece of mozzarella. Save the remaining sauce for another use. Top the mozzarella with Parmesan cheese and place in the oven.
Bake chicken for 15-18 minutes, then turn the broiler on low and broil for about 2 minutes, to brown the cheese.
To serve, top chicken with olive and tomatoes and enjoy!