01 - Bring a large pot of generously 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 have softened and the onions turn translucent.
03 - Stir in the minced garlic and cook for 1 minute, stirring constantly to prevent burning.
04 - Sprinkle the all-purpose flour evenly over the softened vegetables. Stir constantly for 1 to 2 minutes, allowing the flour to cook and coat the vegetables to form a light roux.
05 - Slowly pour in the chicken broth, whole milk, and heavy cream while whisking vigorously to prevent lumps. Continue stirring until the mixture is completely smooth. Bring to a gentle simmer over medium heat.
06 - Season the sauce with dried thyme, dried parsley, salt, and freshly ground black pepper. Fold in the frozen peas and cubed cooked chicken. Maintain a gentle simmer for 4 to 5 minutes until the peas are tender, the chicken is heated through, and the sauce has slightly thickened.
07 - Add the drained pasta to the skillet, tossing and folding until every piece is evenly coated in the creamy sauce. Taste and adjust seasoning with additional salt and pepper as needed.
08 - Preheat the broiler. Transfer the pasta mixture to an oven-safe baking dish. In a small bowl, combine the cracker crumbs, grated Parmesan cheese, and melted butter until the mixture resembles wet sand. Sprinkle evenly over the pasta. Broil for 2 to 3 minutes, watching closely, until the topping turns a deep golden brown.
09 - Serve immediately while hot, garnished with additional chopped fresh parsley if desired.