Sauté the onions and garlic: In a large skillet, heat the olive oil or butter over medium heat. Add the finely chopped onion and sauté for 3-4 minutes until softened and translucent. Incorporate the minced garlic and sauté for an additional 1 to 2 minutes, allowing it to release its aromatic flavor.

Cook the mushrooms: Add the sliced mushrooms to the skillet and cook for 5-6 minutes, stirring occasionally, until they release their moisture and start to turn golden brown.

Blanch the broccoli: While the mushrooms are cooking, blanch the broccoli florets in a pot of boiling water for 2-3 minutes, just until they are bright green and slightly tender. Drain and set aside.

Make the cream sauce: Once the mushrooms are golden, pour in the heavy cream (or half-and-half for a lighter option) and stir in the grated Parmesan cheese. Add the thyme, nutmeg (if using), salt, and freshly ground black pepper to taste. Allow the sauce to gently bubble for 2 to 3 minutes, stirring occasionally until it begins to thicken.

Add the broccoli: Stir the blanched broccoli into the creamy mushroom sauce, making sure the florets are evenly coated in the sauce. Let it cook for another 2-3 minutes, just until the broccoli is warmed through.

Serve: Remove the skillet from heat and sprinkle with fresh parsley before serving. Enjoy this creamy delight on its own, or pair it with rice, pasta, or crusty bread for a complete meal.