Steam the broccoli until tender, or add it to the tortellini pot for the last 2–3 minutes of cooking; drain the tortellini and reserve 1/4 cup of the pasta cooking liquid before draining.
While the pasta cooks, melt the butter in a large skillet over medium heat, then add the chopped onion and cook until softened, about 2 minutes.
Add the minced garlic, red pepper flakes, and a pinch of nutmeg to the skillet and cook, stirring, for about 2 minutes until fragrant.
Pour in the half-and-half and bring the mixture to a gentle simmer; let it simmer for 2–3 minutes to slightly thicken.
Stir in the grated Parmesan and reduce the heat to low; once the cheese is melted, add the drained tortellini, cooked broccoli, and up to 1/4 cup reserved pasta water, tossing to coat and thicken the sauce.
Season to taste with salt and pepper, heat a few minutes more if needed, and serve with additional Parmesan.