Preheat oven to 350 degrees. Butter a 2-quart baking dish. In a medium microwave-safe bowl, combine carrots with 3 tablespoons water. Cover with plastic wrap and microwave until tender, 4 minutes. Drain carrots in a fine-mesh sieve. In a large bowl, whisk together flour, baking powder, baking soda, and salt. Whisk in eggs, sugar, and butter until combined. Fold in carrots and transfer mixture to prepared dish. Bake until top is golden brown, edges pull away from sides of dish, and a toothpick inserted in center comes out clean, 55 to 60 minutes. Let cool slightly before serving.