Preheat the oven to 350 degrees F (175 degrees C). Lightly grease a 2-quart casserole dish. Bring a large pot of salted water to a boil. Add carrots and cook until tender, 15 to 20 minutes. Drain and mash. Stir in butter, vanilla extract, and eggs; mix well. Sift flour, baking powder, salt, and sugar…