Preheat the oven to 375°F (190°C). Grease a 9x13-inch baking dish or similar casserole pan.
Warm a large skillet over medium heat. Add butter and olive oil.
Sauté the onion for 3–4 minutes until softened.
Add the garlic and mushrooms. Cook, stirring occasionally, until the mushrooms release liquid and it evaporates, about 6–8 minutes. They should be golden and concentrated in flavor.
Stir in the thyme, smoked paprika, salt, and pepper.
If using greens, add spinach or kale and cook 1–2 minutes until wilted.
Lower the heat. Add the cream cheese and stir until it melts into the mushrooms.
Pour in the broth, heavy cream, and Dijon. Simmer gently for 2–3 minutes, stirring, until the sauce is smooth and slightly thickened.
Fold in the cooked turkey and half the mozzarella.
Taste and adjust salt and pepper as needed.
Transfer the mixture to the baking dish. Top with remaining mozzarella and the Parmesan.
Bake for 15–18 minutes, until bubbly around the edges and lightly browned on top. If you want more color, broil for 1–2 minutes at the end.
Let it rest for 5–10 minutes.
Garnish with parsley and serve warm.