Mix the flour and some salt and pepper together on a large plate. Dredge the flattened chicken breasts in the flour mixture. Set aside.
Cook the linguine until al dente.
Heat the olive oil and butter together in a large skillet over medium heat. When the butter is melted and the oil/butter mixture is hot, fry the chicken breasts until nice and golden brown on each side, 2 to 3 minutes per side. Remove the chicken breasts from the skillet and set aside.
Without cleaning the skillet, add the garlic and onions and gently stir for 2 minutes. Pour in the stock and scrape the bottom of the pan, getting all the flavorful bits off the bottom. Allow the stock to cook down until reduced by half, about 2 minutes.
Pour in the crushed tomatoes and stir to combine. Add the sugar and more salt and pepper to taste. Allow to cook for 30 minutes.
Toward the end of the cooking time, add the chopped parsley and give the sauce a final stir.
Carefully lay the chicken breasts on top of the sauce and completely cover them in the grated Parmesan. Place the lid on the skillet and reduce the heat to low. Allow to simmer until the cheese is melted and the chicken is thoroughly heated. Add more cheese to taste.
Place the cooked noodles on a plate and cover with the sauce. Place the chicken breasts on top and sprinkle with more parsley and the basil chiffonade. Serve immediately.