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 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 and the onions turn translucent.
03 - Stir in the minced garlic and cook for 1 minute, stirring constantly to prevent burning, until fragrant.
04 - Sprinkle the all-purpose flour evenly over the sautéed vegetables. Stir constantly for 1 to 2 minutes, coating the vegetables thoroughly, 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 continuously to prevent lumps. Continue whisking 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, stirring occasionally, 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 evenly coated in the creamy sauce. Taste and adjust salt and pepper as needed.
08 - Preheat your oven 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. Sprinkle the mixture evenly over the top. Broil for 2 to 3 minutes, watching carefully, until the topping is golden brown and bubbling.
09 - Serve immediately while hot, garnished with additional chopped fresh parsley if desired.