01 - Bring a large pot of well-salted water to a rolling boil. Add the pasta and cook according to package directions until al dente. Drain in a colander and set aside.
02 - While the pasta cooks, melt 3 tbsp 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 onions are translucent.
03 - Stir in the minced garlic and cook for 1 minute until fragrant, being careful not to let it brown.
04 - Sprinkle the flour evenly over the sautéed vegetables. Stir constantly for 1 to 2 minutes to cook out the raw flour taste and form a light roux.
05 - Slowly whisk in the chicken broth, followed by the whole milk and heavy cream, stirring until the mixture is smooth and free of lumps. Bring to a gentle simmer over medium heat.
06 - Season the sauce with dried thyme, dried parsley, salt, and freshly ground black pepper. Stir in the frozen peas and cubed cooked chicken. Simmer for 4 to 5 minutes until the chicken is heated through and the sauce has thickened slightly.
07 - Add the drained pasta to the skillet, folding gently until every piece is well 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, and melted butter. Sprinkle evenly over the pasta and broil for 2 to 3 minutes until the topping is golden brown and bubbly.
09 - Serve immediately while hot, garnished with additional chopped fresh parsley if desired.