Set a large pot of salted water to boil and preheat oven to 400°F.
While waiting for the water to boil, prep the broccoli by cutting it into florets, then tossing with 1 teaspoon olive oil and a bit of salt and pepper. Spread on a baking sheet and roast for 15-20 minutes.
Cook pasta according to package instructions to al dente, reserve ½ cup pasta water, and drain pasta.
While pasta is cooking, cook chicken. Pat chicken dry with a paper towel and season with salt, pepper, oregano, and thyme.
Heat 1 tablespoon olive oil in a pan over medium-high. Cook chicken 5-8 minutes per side (or until done). Slice thinly and keep warm.
Deglaze pan with wine or broth, add garlic, and simmer to reduce slightly, lowering heat to medium.
Stir in cream, mozzarella, parmesan, salt, pepper, and red pepper flakes. Whisk to heat through and melt the cheese until it is completely smooth. Reduce heat to low.
Add pasta and toss to coat, adding pasta water if the sauce seems too thick.
Add chicken and broccoli, tossing to coat. Serve with extra parmesan cheese, red pepper flakes, and parsley, if desired.