Bring a large pot of well-salted water to a boil. Cook the refrigerated tortellini according to the package directions, usually 3–5 minutes, then reserve about 1/2 cup pasta water and drain.

Meanwhile, heat the butter and olive oil in a large skillet over medium-high heat. Add the sliced mushrooms and cook for 5–7 minutes, stirring occasionally, until browned and most of their moisture has evaporated.

Add the diced onion and cook for 2–3 minutes, until softened. Stir in the minced garlic and cook for about 30 seconds, just until fragrant.

Sprinkle the flour over the mushroom mixture and stir continuously for 1 minute. This cooks away the raw flour taste and prepares the base for a smooth sauce.

Gradually whisk in the broth, followed by the heavy cream. Stir in the Italian seasoning, kosher salt, and black pepper, then simmer for 3–5 minutes until slightly thickened.

Stir in the finely grated Parmesan until melted. Add the drained tortellini and toss gently to coat, loosening the sauce with a splash of reserved pasta water if necessary.

Finish with chopped parsley and additional Parmesan if desired. Serve immediately while the sauce is warm and glossy.