Adjust the oven rack to the middle position and preheat the oven to 425°F (218°C).
Heat 1 tablespoon of the oil reserved from the sun-dried tomato jar in a large 3.5‑quart oven-safe skillet over medium-high heat.
Add the sliced mushrooms and cook 2–3 minutes until they begin to soften, then add the sun-dried tomatoes and cook another 2–3 minutes.
Add the minced garlic and cook 30 seconds until fragrant, then transfer the vegetables and any accumulated liquid to a plate and set aside.
In the same skillet, melt the remaining 4 tablespoons of butter over medium heat. Whisk in the flour and cook, whisking constantly, 1–2 minutes until smooth and slightly bubbly.
Stir in the Italian seasoning, paprika, red pepper flakes (if using), and salt and pepper to taste.
While whisking constantly, slowly pour in the milk and chicken broth until the sauce is smooth. Bring to a gentle 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, adding about 2 tablespoons at a time and stirring until melted before adding more.
Return the cooked mushrooms and sun-dried tomatoes (with any juices) to the sauce. Add the shredded rotisserie chicken, refrigerated tortellini, and chopped spinach. Gently stir to combine and press the tortellini into the sauce so they are mostly submerged.
Transfer the skillet to the preheated oven and bake for 8 minutes.
Remove from the oven, stir to even out the mixture, press into an even layer, sprinkle the remaining 3 tablespoons Parmesan over the top, and return to the oven for an additional 3 minutes.
Remove from the oven, taste and adjust salt and pepper as needed, and serve immediately.
