Heat the oven. Preheat to 375°F (190°C). Grease a 9x13-inch baking dish and set aside.
Par-cook the tortellini. Bring a large pot of salted water to a boil.
Cook the tortellini 1–2 minutes less than package directions. Drain well. This prevents overcooking in the oven.
Sauté aromatics. In a large skillet, warm olive oil or butter over medium heat.
Add onion and a pinch of salt. Cook 4–5 minutes until translucent. Stir in garlic for 30 seconds until fragrant.
Make the roux. Sprinkle flour over the onion mixture.
Stir for 1 minute to cook off the raw flour taste.
Build the sauce. Slowly whisk in chicken broth until smooth, then add cream. Simmer gently 3–4 minutes, stirring, until slightly thickened.
Season and enrich. Add Italian seasoning, red pepper flakes (if using), 1/2 teaspoon salt, and several grinds of pepper. If using cream cheese, add it now and stir until melted and silky.
Add greens and chicken. Fold in spinach and cook 1–2 minutes until wilted.
Stir in the cooked chicken. Taste and adjust seasoning. The sauce should be well-salted and flavorful.
Combine with tortellini. Add drained tortellini to the skillet and gently fold to coat every piece in sauce.
Assemble the casserole. Transfer to the prepared baking dish.
Top evenly with mozzarella and Parmesan.
Bake. Bake uncovered 18–22 minutes, until bubbly at the edges and the cheese is melted.
Broil for color (optional). Switch to broil for 1–2 minutes to brown the top slightly. Watch closely.
Rest and garnish. Let sit 5 minutes to set. Sprinkle with parsley or basil and serve warm.