Preheat oven to 350°F. Spray a 9 x 9-inch baking dish (or other 2 - 2 ½ -quart baking dish) with cooking spray and set aside.
Cook the noodles in a large pot of salted boiling water according to package directions, just until al dente. Drain.
Meanwhile, melt 1 tablespoon of the butter in a large skillet or Dutch oven over medium-high heat. Add the carrots, celery, onion, and garlic; sauté until soft (about 5-7 minutes).
Reduce the heat to medium. Add 3 tablespoons of butter to the skillet and stir until the butter melts. Add the flour and whisk until blended. Gradually add the milk and the chicken broth, whisking constantly. Continue whisking and cooking until the sauce bubbles and thickens. Stir in the thyme and season with salt and pepper to taste. Remove from heat.
In a large bowl (or in the same pot if it’s large enough), combine the sauce with the cooked noodles, chicken, cheese, and frozen peas. Transfer to the prepared dish.
Melt the remaining 2 tablespoons of butter. Combine the melted butter with the Ritz cracker crumbs. Sprinkle buttered crumbs over top of the casserole.
Bake the casserole (uncovered) for about 20-25 minutes, or until hot and bubbly.