Adjust the oven rack to the middle position and preheat oven to 425°F.
In a large (3.5‑quart) oven-safe skillet over medium-high heat, add 1 tablespoon oil reserved from the sun-dried tomato jar. Sauté the sliced mushrooms 2–3 minutes until they begin to soften.
Add the sun-dried tomatoes to the pan and cook 2–3 minutes more, then stir in the minced garlic and cook 30 seconds; transfer the vegetables and any accumulated liquid to a plate.
Lower heat to medium and melt the remaining 4 tablespoons butter in the same pan. Sprinkle the flour over the melted butter and whisk continuously until smooth, 1–2 minutes.
Whisk in the Italian seasoning, paprika, red pepper flakes (if using), and season with salt and pepper.
While whisking constantly, slowly pour in the milk and chicken broth until the sauce is smooth. Bring to a boil, then reduce to medium and simmer 3–5 minutes until slightly thickened.
Reduce heat to low and add the grated Parmesan in 2-tablespoon increments, stirring until melted and smooth before adding more.
Return the cooked mushrooms, sun-dried tomatoes, and any juices to the sauce. Add the shredded chicken, refrigerated tortellini, and chopped spinach; gently stir to combine and press the tortellini into an even layer so it is mostly submerged in the sauce.
Transfer the skillet to the preheated oven and bake 8 minutes. Remove, stir and press into an even layer, then sprinkle the remaining Parmesan over the top and bake 3 more minutes.
Remove from oven, taste and adjust salt and pepper as needed, then serve immediately.