01 - Bring a large pot of salted water to a rolling boil. Add the pasta and cook according to package directions until al dente. Drain through a colander and set aside.
02 - While the pasta cooks, melt 3 tbsp unsalted butter in a large deep skillet or Dutch oven over medium heat. Add the diced onion, carrots, and celery. Sauté for 5 to 7 minutes, stirring occasionally, until the vegetables are softened and the onion is translucent.
03 - Stir in the minced garlic and cook for 1 minute, stirring constantly to prevent burning.
04 - Sprinkle the flour evenly over the sautéed vegetables. Stir constantly for 1 to 2 minutes, allowing the flour to cook and form a light roux that coats the vegetables.
05 - Slowly pour in the chicken broth, whole milk, and heavy cream while whisking continuously to prevent lumps. Stir until the mixture is completely smooth, then bring to a gentle simmer.
06 - Season the sauce with dried thyme, dried parsley, salt, and freshly ground black pepper. Fold in the frozen peas and cubed cooked chicken. Simmer for 4 to 5 minutes until the chicken is heated through and the sauce has slightly thickened.
07 - Add the drained pasta to the skillet, folding gently until every piece is evenly coated in the creamy sauce. Taste and adjust salt and pepper as needed.
08 - If using the topping, preheat the broiler. Transfer the pasta mixture to an oven-safe baking dish. Combine the cracker crumbs, grated Parmesan, and melted butter in a small bowl, then sprinkle evenly over the pasta. Broil for 2 to 3 minutes until the topping is golden brown and bubbly.
09 - Serve hot, garnished with additional fresh or dried parsley if desired.