Preheat oven to 350 degrees F. Generously butter a square 8-inch baking dish.
Peel the carrots and trim the ends. Cut each carrot in half. Place in a large stockpot. Fill with cold water and bring to a rolling boil, then lower to gentle boil and cook the carrots are until very tender, about 20 minutes.
Meanwhile, in your food processor bowl, add the remaining ingredients and pulse to combine.
Drain the carrots; allow them to slightly cool, then gradually add them to the food processor, continuously processing the mixture until light and fluffy.
Using a rubber spatula, transfer the carrot mixture to the prepared baking dish.
Bake, uncovered, until edges are brown and a toothpick inserted in center comes out dry, about 50 minutes.
Cool in pan on a wire rack for 10 minutes, then carefully run a knife around the edges to loosen, cut into four squares and serve.