Adjust the oven rack to the middle and preheat to 425°F.
In a large (3.5 quart) oven-safe pan over medium-high heat, heat 1 tbsp oil from the sun-dried tomatoes. Add mushrooms and cook for 2–3 minutes. Then add sun-dried tomatoes and cook another 2–3 minutes. Add minced garlic and cook for 30 seconds. Transfer everything to a separate plate.
In the same pan, melt remaining 4 tbsp butter. Sprinkle flour over the melted butter and whisk briskly until smooth. Continue cooking and whisking constantly for 1–2 minutes. Stir in Italian seasoning, paprika, red pepper flakes (if using), plus salt and pepper to taste.
While whisking constantly, very gradually pour in the milk and chicken broth until the mixture is smooth. Bring to a boil, then reduce heat to medium and simmer until slightly thickened, about 3 to 5 minutes. Reduce heat to low and gradually add 2 tablespoons of Parmesan cheese at a time, stirring until melted.
Return the cooked veggies (along with accumulated liquid) to the pan. Add rotisserie chicken, tortellini, and chopped spinach. Gently stir to combine, ensuring the tortellini is submerged in the liquid as much as possible and smooth into an even layer.
Place pan in oven and bake for 8 minutes. Remove from oven, stir everything well, and press into an even layer. Sprinkle the remaining 3 tablespoons of Parmesan over the top and bake for an additional 3 minutes.
Remove from oven, season with additional salt and pepper if needed, and garnish with fresh herbs if desired. Serve immediately for best results, as the tortellini will continue to absorb the sauce if left to sit.