Fill a stock pot with water and bring to a boil. Add the egg noodles and cook until al dente (about 2 minutes less than package directions). Drain and pour the noodles into a 9 x 13 casserole dish sprayed with nonstick cooking spray.
In a large skillet on medium heat, melt butter. Add onion and celery and sauté until tender. Stir in minced garlic and flour and cook 1 minute. Then slowly pour in chicken broth and milk, season with salt and pepper, and bring to a simmer, stirring constantly.
Once mixture has thickened, stir in mixed vegetables, shredded chicken, and cheddar cheese. Remove from heat and stir until cheese has melted. Pour the mixture over the egg noodles in the casserole dish.
Top with shredded cheddar cheese, then Ritz crackers, parmesan cheese, and drizzle with melted butter. Bake uncovered for 15 minutes, or until crackers are golden brown.