FOR the Basic Tomato Sauce:

1/4 cup extra-virgin olive oil
1 3/4 cups chopped onions
4 garlic cloves, minced
3/4 cup coarsely grated peeled carrots
3 tablespoons chopped fresh thyme
2 28-ounce cans peeled whole tomatoes in juice

FOR the Chicken:

10 skinless boneless chicken breast halves
3 cups fresh breadcrumbs (from crustless French bread ground in processor)
2 large eggs
1 cup (about) all purpose flour
6 tablespoons (or more) olive oil, divided
3 cups coarsely grated well-drained fresh water-packed mozzarella,* divided
1 1/4 cups freshly grated Parmesan cheese, divided
1 1/4 cups freshly grated Pecorino Romano cheese, divided
2 tablespoons chopped fresh Italian parsley
1 tablespoon chopped fresh marjoram