Heat oven to 375°F and lightly grease a 9x13-inch baking dish.
Season chicken with 1 tsp salt, pepper, Italian seasoning, and smoked paprika.
Warm olive oil in a large skillet over medium-high heat. Sear chicken until lightly browned, 4 to 5 minutes. Transfer to a plate.
Melt butter in the same skillet. Cook onion and mushrooms until tender and lightly golden, 4 to 5 minutes.
Stir in garlic and flour. Cook, stirring, for 30 seconds to remove raw flour taste.
Whisk in chicken broth. Simmer until slightly thickened, 2 to 3 minutes.
Reduce heat to low. Add cream cheese and heavy cream, stirring until smooth and creamy.
Fold in spinach until wilted. Season sauce with remaining salt and red pepper flakes to taste.
Return chicken with any juices to the skillet. Stir to coat evenly in the sauce.
Spread mixture in the prepared baking dish. Sprinkle Parmesan and top with mozzarella.
Bake until bubbly and chicken is cooked through, 18 to 22 minutes. Broil 1 to 2 minutes for extra browning if desired.
Rest 5 minutes. Finish with parsley and serve warm.