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 tablespoons of 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.
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 softened vegetables. Stir constantly for 1 to 2 minutes to cook out the raw flour taste and form a light roux.
05 - Slowly pour in the chicken broth, whole milk, and heavy cream, whisking constantly to prevent lumps. Continue stirring until the mixture is smooth, then bring to a gentle simmer.
06 - Season the sauce with dried thyme, dried parsley, salt, and pepper. Fold in the frozen peas and cubed chicken. Let the mixture simmer for 4 to 5 minutes until the peas are tender, the chicken is heated through, and the sauce has thickened slightly.
07 - Add the drained pasta to the skillet and toss until every piece is well coated in the creamy sauce. Taste and adjust 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 top. Broil for 2 to 3 minutes until the topping is golden brown and bubbling.
09 - Serve hot, garnished with fresh parsley if desired.