Place carrots and onions in a pot with enough lightly salted water to cover, and bring to a boil. Cook 5 minutes, or until tender but firm. Drain, and spread 1/2 the carrots and onions in a baking dish. Cover with 3/4 cup cheese, layer with remaining carrots and onions, and top with remaining cheese.
In a saucepan, melt 3 tablespoons butter and mix in flour until smooth. Gradually stir in milk until thickened, and season with salt and pepper. Pour over the carrots, onions, and cheese in the baking dish.
Melt 1 tablespoon butter in a small pot, and mix in bread crumbs. Sprinkle evenly in the baking dish.
Bake 20 minutes in the preheated oven, or until bubbly and lightly browned.