Bring a large pot of salted water to a boil. Cook pasta according to package directions until al dente. Reserve 1 cup of pasta water before draining. Drain pasta and set aside.

In a large pot or Dutch oven, heat olive oil and 1 tbsp butter over medium heat. Add onion, carrots, and celery. Sauté for 6-8 minutes until softened. Add garlic and cook for 1 more minute.

Add remaining 2 tbsp butter to the pot. Once melted, sprinkle flour over vegetables. Cook, stirring constantly, for 1-2 minutes to form a roux and cook off the raw flour taste.

Gradually whisk in the chicken broth, ensuring no lumps form. The sauce will thicken. Whisk in the milk and heavy cream until smooth.

Bring sauce to a gentle simmer. Stir in dried thyme and cooked chicken. Simmer for 3-5 minutes to meld flavors and thicken sauce. Stir in frozen peas.

Season the sauce generously with salt and pepper. Add the cooked pasta to the pot and stir to combine. If sauce is too thick, add reserved pasta water a splash at a time until desired consistency is reached.

Garnish with fresh parsley and serve immediately.