Boil the tortellini: Bring a large pot of salted water to a boil.
Cook tortellini according to package directions until just al dente. Reserve 1/2 cup pasta water, then drain. Toss with a drizzle of olive oil to prevent sticking.
Season the chicken: Pat the chicken dry and season with 1 teaspoon salt and 1/2 teaspoon black pepper.
Add a pinch of red pepper flakes if you like heat.
Sear the chicken: Heat 1 tablespoon olive oil in a large skillet over medium-high. Add chicken in a single layer and cook, undisturbed, until lightly browned, 3–4 minutes. Flip and cook through, another 3–4 minutes.
Transfer to a plate.
Brown the mushrooms: In the same skillet, add 1 tablespoon butter. Add mushrooms with a pinch of salt. Cook without stirring for 2 minutes to get color, then sauté until golden and their liquid evaporates, 5–7 minutes.
Sauté aromatics: Push mushrooms to the side.
Add remaining butter and the shallot. Cook 1–2 minutes until softened. Stir in garlic and cook 30 seconds until fragrant.
Deglaze: Pour in white wine (if using).
Scrape up browned bits. Simmer 1–2 minutes to reduce by half.
Build the sauce: Add chicken broth and bring to a gentle simmer. Reduce heat to medium-low.
Stir in heavy cream and half the Parmesan. Simmer 2–3 minutes until slightly thickened.
Combine: Return chicken and any juices to the pan. Fold in the cooked tortellini.
If sauce is too thick, add a splash of reserved pasta water. If too thin, simmer 1–2 minutes to reduce.
Finish with brightness: Add lemon zest and a squeeze of lemon juice to taste. Stir in herbs.
Season with salt and pepper as needed.
Serve: Top with remaining Parmesan and extra herbs. Serve hot with a simple green salad or steamed vegetables.