Adjust oven rack to the middle position and preheat the oven to 425°F (218°C).
In a large 3.5‑quart oven‑safe skillet over medium-high heat, add 1 tablespoon oil reserved from the sun-dried tomatoes and heat until shimmering.
Add the sliced mushrooms and cook 2–3 minutes until they begin to brown, then add the sun-dried tomatoes and cook another 2–3 minutes.
Stir in the minced garlic and cook about 30 seconds until fragrant, then transfer the cooked vegetables and any accumulated juices to a small plate and set aside.
Reduce heat to medium and melt the remaining 4 tablespoons butter in the same skillet. Sprinkle the flour over the melted butter and whisk briskly until smooth, cooking 1–2 minutes while whisking constantly.
Add the Italian seasoning, paprika, red pepper flakes (if using), and salt and pepper to taste, stirring to combine.
While whisking constantly, slowly pour in the milk and chicken broth until smooth. Bring the mixture to a boil, then reduce heat to medium and simmer 3–5 minutes until slightly thickened.
Reduce heat to low and gradually stir in the Parmesan cheese, about 2 tablespoons at a time, stirring until each addition melts and the sauce is smooth.
Return the cooked mushrooms and sun-dried tomatoes (with any juices) to the skillet. Add the shredded rotisserie chicken, refrigerated tortellini, and chopped spinach; gently stir to combine and ensure the tortellini is submerged in the sauce.
Place the skillet in the preheated oven and bake 8 minutes. Remove, stir well and press into an even layer, then sprinkle the remaining 3 tablespoons Parmesan over the top.
Return to the oven and bake an additional 3 minutes until the top is lightly browned and the sauce is bubbly. Remove, adjust seasoning with salt and pepper if needed, and serve immediately.