Heat the oven to 375°F, grease a 9-by-13-inch baking dish and boil the noodles 3 minutes short of tender.
Melt the butter in a skillet and cook the mushrooms and onion until browned and their liquid evaporates, 8 to 10 minutes.
Whisk the condensed soup, sour cream, broth, salt and black pepper in a large bowl.
Fold in the drained noodles, mushroom mixture, cooked chicken and frozen peas and spread everything in the dish.
Scatter the crushed crackers over the top and bake until bubbling and browned, 25 to 30 minutes.
Rest for 8 minutes before serving so the sauce settles around the noodles.