Season chicken with salt, pepper, paprika, parsley, and 2 tsp olive oil. Sear in 1 Tbsp oil over medium-high heat for 6–8 minutes until golden and cooked through (165°F). Set aside, covered.
Lower heat to medium. Melt butter in the same pan, add onion and garlic, and cook until soft—about 3 minutes.
Pour in wine, scrape the pan, and simmer 2–3 minutes. Stir in sun-dried tomatoes and 2 Tbsp of their oil. Cook 1–2 minutes.
Sprinkle in flour and stir for 1 minute. Gradually whisk in stock and milk until smooth. Add herbs, salt, and pepper. Simmer gently.
Add pasta directly to sauce. Simmer on medium-low, stirring now and then, until tender and thickened—about 9–12 minutes.
Stir in spinach until wilted—1–2 minutes.
Take off heat. Stir in Parmesan, cheddar, and mozzarella. Add a splash of milk if needed to loosen.
Slice or shred chicken and fold into pasta. Stir in parsley and serve warm.