Heat 3 tablespoons of the olive oil in a very large skillet over medium-high heat. Add the mushrooms, season them generously with salt and pepper and cook, making sure not to move around too much, until the mushrooms release their water, 5-6 minutes. Remove the mushrooms to a plate and reserve.
Add the remaining 5 tablespoons oil, then add the onions and cook until golden, 7-8 minutes. Add the garlic and cook 1 additional minute. Add the diced tomatoes and chili flakes, and more salt and pepper to taste, bring to a boil, reduce heat and cook until sauce thickens, 4-5 minutes.
Add the reserved mushrooms to the sauce along with the chicken breast, toss gently, cover and keep warm. Cook the ravioli according to package instructions and drain, reserving ½ cup of the pasta cooking water.
Add the ravioli to the sauce, adding pasta water a little at a time to loosen the sauce if necessary. Toss in the parsley, divide among bowls and garnish with more cheese.