Heat olive oil in large saucepan over medium-high heat.
Add onions and garlic; sauté until onions are soft and golden, about 10 minutes.
Add carrots and thyme; sauté until carrots are soft, about 5 minutes.
Add tomatoes with juice; bring to boil, coarsely crushing tomatoes with potato masher or fork.
Reduce heat to medium-low; simmer until sauce thickens and is reduced to about 2 cups, about 1 hour. Season sauce to taste with salt and pepper.
Make-ahead instructions: Sauce can be made 1 day ahead. Cool slightly after cooking. Chill uncovered until cold, then cover and keep refrigerated. Rewarm sauce before using.
Place chicken breast halves between 2 sheets of plastic wrap. Using meat mallet or rolling pin, pound chicken breasts to ⅓-inch thickness. Sprinkle both sides of chicken with salt and pepper.
Spread breadcrumbs on plate. Whisk egg in medium bowl. Spread flour on another plate.
Coat both sides of chicken with flour, then eggs, then breadcrumbs.
Heat oven to 350 F.
Heat 2 tablespoons oil in large nonstick skillet over medium-high heat.
Add chicken to skillet (in batches, if necessary) and cook until brown, about 2 minutes per side, adding more oil as needed (chicken will not be cooked through). Transfer chicken to platter.
Spread a layer of sauce over bottom of 15x10x2-inch glass baking dish.
Arrange chicken over sauce. Spoon remaining sauce over. Sprinkle mozzarella, Parmesan, and Pecorino on top of chicken. Continue to repeat layers until all chicken is used.
Bake until cheeses melt and chicken is cooked through, about 20 minutes. Sprinkle with parsley and marjoram and serve.