Preheat the oven to 400°F (200°C). Bring a large pot of water to a rolling boil, add the wide egg noodles, and cook them until al dente. Drain thoroughly and place the noodles aside.
Melt 1 stick of unsalted butter in a large skillet set over medium heat. Add the chopped onion and diced celery, then sauté until both vegetables become tender.
Add the minced garlic to the skillet and continue cooking for approximately 1 minute, just until fragrant.
Sprinkle the flour over the cooked vegetables and whisk until fully incorporated. Cook the mixture for 1–2 minutes to remove the raw taste of the flour.
Slowly add the chicken broth while stirring continuously. Continue mixing until the sauce begins to thicken.
Pour in the whole milk and stir until evenly blended. Season the sauce with the salt and black pepper.
Fold the mixed vegetables and cooked chicken into the sauce. Remove the skillet from the heat, then stir in the shredded sharp cheddar, setting aside a portion of the cheese for the casserole topping.
Lightly grease a 9 × 13-inch baking dish. Add the cooked egg noodles and chicken mixture, stirring them together until the noodles are evenly coated and the ingredients are well distributed.
Scatter the reserved cheddar cheese evenly across the surface of the casserole.
Combine the crushed Ritz crackers with the grated Parmesan cheese. Distribute the cracker mixture evenly over the casserole, then drizzle the melted 1/2 stick of butter across the topping.
Bake in the preheated oven for 15–20 minutes, or until the topping develops a crisp, golden-brown color.
Remove the casserole from the oven and allow it to cool briefly before serving.