Set the crockpot to high. In a skillet or directly in the crockpot if it allows, melt the butter and stir in the flour to make a roux; cook for about 3–5 minutes until slightly golden.
Add the chopped onion, garlic, Italian seasoning, and chili flakes to the roux and stir until the onion begins to soften, about 2–3 minutes.
Stir in the chopped celery, drained and chopped sun-dried tomatoes, and the broth. Season with salt and black pepper to taste.
Cover and cook in the crockpot on low for 4–6 hours or on high for 1–2 hours (recipe uses crockpot instructions).
During the final 15–20 minutes of cooking, increase the crockpot to high (if not already). Stir in the pasta, chopped spinach, heavy cream, and milk. Cook on high until the pasta is tender, about 20 minutes; alternatively, cook the pasta separately in boiling water and add it to the soup.
Stir in the shredded provolone and grated Parmesan until melted and incorporated; thin with additional broth or water if the soup is too thick.
Ladle into bowls and serve warm.