Bring a large pot of salted water to a boil. Add pappardelle and cook according to package instructions until al dente, about 7-8 minutes.
Reserve ½ cup pasta water, then drain the pasta and set aside.
Meanwhile, heat olive oil in a large, deep skillet over medium-high heat. Season chicken with ½ teaspoon salt and ¼ teaspoon black pepper. Add chicken and cook until golden and cooked through, 5-6 minutes. Remove to a plate and set aside.
In the same skillet, melt the butter. Add onion, carrots, and celery with remaining salt and pepper. Sauté until veggies are soft and aromatic, 5-6 minutes. Add garlic and cook for 1 more minute.
Sprinkle flour over the veggies and stir, cooking for 1-2 minutes until the flour is fully incorporated and lightly golden.
Gradually whisk in chicken broth, then milk and heavy cream, ensuring there are no lumps.
Add thyme, dried parsley, and smoked paprika (if using). Simmer for 4-5 minutes until the sauce thickens.
Return chicken to the pan. Stir in frozen peas and Parmesan cheese. Simmer another 2-3 minutes until heated through and creamy. Adjust salt and pepper to taste.
Add drained pappardelle to the skillet, tossing gently to coat in the sauce. If it’s too thick, add some reserved pasta water as needed.
Divide into bowls and sprinkle generously with fresh parsley. Serve immediately for maximum creaminess!