Bring a large pot of salted water to a boil and cook the egg noodles 1–2 minutes less than the package directions — they should be just shy of al dente, since they'll finish cooking in the oven. Drain and set aside. (Slightly undercooked noodles won't turn mushy in the creamy sauce.)

In a large skillet or Dutch oven, melt the 4 tablespoons of butter over medium heat. Whisk in the flour and cook, whisking constantly, for 1 minute to cook out the raw flour taste. Slowly whisk in the chicken broth, then the milk, and bring to a gentle simmer, whisking, until the sauce thickens enough to coat the back of a spoon (3–4 minutes). Remove from the heat and whisk in the sour cream, garlic powder, onion powder, salt, and black pepper until smooth.

Fold the shredded chicken, drained noodles, thawed mixed vegetables, and 1½ cups of the cheddar into the sauce until everything is evenly coated. Spread the mixture into a greased 9×13-inch baking dish in an even layer.

In a small bowl, toss the crushed crackers with the melted butter until evenly moistened. Sprinkle the buttered crumbs over the casserole, then scatter the remaining ½ cup of cheddar over the top.

Bake at 350°F (175°C) for 25–30 minutes, until the sauce is bubbling around the edges and the topping is deep golden brown. Let it rest for 5 minutes before serving so the sauce sets up.