Heat oven to 400°F. Pat chicken dry and season with 1 tsp salt, pepper, paprika, and Italian seasoning.
Warm olive oil and 1 tbsp butter in a large oven-safe skillet over medium-high heat.
Sear chicken until deeply golden, 3–4 minutes per side. Transfer to a plate; keep warm.
Add remaining butter to the skillet. Cook mushrooms with remaining 0.25 tsp salt until browned, 4–5 minutes.
Stir in garlic and red pepper flakes; cook 30 seconds until fragrant.
Sprinkle flour over mushrooms and cook 30 seconds, stirring to coat.
Whisk in broth and then cream, scraping up browned bits. Simmer 2–3 minutes to thicken slightly.
Blend in Parmesan. Return chicken and juices to the pan, spooning sauce over the pieces.
Scatter mozzarella on top. Bake until cheese melts and chicken reaches 165°F, 8–10 minutes.
Finish with parsley and lemon juice. Rest 5 minutes, then serve with the creamy mushroom sauce.