For the sauce: Heat 2 tablespoons olive oil in a medium skillet over medium heat. Add the onion and a pinch of salt and cook for a few minutes until the onions are softened slightly but not brown. Stir in the garlic, sugar and red pepper flakes. (If the onions begin to brown, add a splash of water.)
Stir in the grated carrots and the remaining 2 tablespoons olive oil and continue to cook until the carrots soften, about 2 minutes. Add the canned tomatoes and use a wooden spoon to break up some of the whole tomatoes. Cook over medium heat, stirring from time to time, until the tomatoes are fairly broken down and the sauce has thickened, at least 30 minutes. Season to taste and add water as necessary. Stir in the basil leaves. Keep the sauce warm.
For the chicken: Lay a piece of chicken between two pieces of plastic wrap and use the smooth side of a meat mallet to pound into a 1/4-inch-thick cutlet. Repeat with remaining cutlet then slice both pieces in half.
Line a baking sheet with paper towels and set aside. Combine the breadcrumbs and garlic powder in a shallow dish. Crack the eggs in a second shallow dish and whisk to blend. Sprinkle the chicken pieces on both sides with salt and dip each thoroughly in the eggs; transfer to the breadcrumbs to thoroughly coat. Place the chicken on the paper towels.
For the pasta: Bring a large pot of water to a rolling boil. Add a generous amount of salt. (The pasta water should taste like seawater.) Add the pasta to the pot and stir so it doesn't stick. Cook the pasta according to the package directions until al dente (chewy but not hard or raw tasting). Drain well and set aside.
To finish the chicken: Fill a medium cast-iron skillet with about 1/3 to 1/2 cup canola oil (to come about 1/4 of the way up the sides) and heat over medium until the oil is just barely starting to smoke. Sprinkle the top of each cutlet with salt and gently place in the oil, salted-side down. Add the butter to the pan and as it begins to foam, shake the pan gently. Cook on the first side until golden brown, 3 to 4 minutes. Gently flip to the second side, sprinkle with salt and carefully spoon some of the hot oil mixture on top. Cook until golden brown and cooked through, an additional 2 to 3 minutes, then remove from the pan with a slotted spatula to clean paper towels to drain slightly.
Preheat the broiler with the rack in the highest position. Divide the sauce between 2 medium bowls, one for the pasta and one for the chicken Parmesan.
To assemble the chicken Parmesan, add some tomato sauce to a small cast-iron skillet or baking dish. Layer with a couple mozzarella slices and the basil. Put one of the fried chicken cutlets on top of the basil, followed by another spoonful of sauce, then a second layer of mozzarella. Place the second chicken cutlet next, followed by a third layer of sauce, more mozzarella and a large sprinkle of Parmesan. Repeat with the remaining pieces of chicken and other ingredients in a second small cast-iron skillet. Place the skillets on a sheet tray and broil until the cheese is melted and slightly browned, 4 to 5 minutes. Top with more freshly grated Parmesan.
Toss the pasta and sauce together in a large bowl and taste for seasoning. Top with grated Parmesan. Serve with the skillets of chicken parm.