Preheat oven to 350°F. Grease a 9×9-inch baking dish and set aside.
Add carrots to large pot of water. Bring to a boil and boil the carrots until very tender (approximately 20 minutes). Drain well and add to a large mixing bowl or the bowl of a stand mixer.
Using a stand mixer or a hand mixer on medium speed, beat the cooked carrots until their texture begins to resemble purée.
Add butter and sugar to the puréed carrots and beat until smooth.
Add eggs one at a time, then the vanilla extract, mixing after each until just combined.
In a separate medium bowl, whisk together flour, baking powder, cinnamon, nutmeg, and salt. Fold in the flour mixture and mix until combined.
Pour the carrot mixture into the prepared baking dish. Bake for 55-60 minutes until lightly browned and the center is no longer liquid.
Let stand for 10 minutes, and if desired, dust with powdered sugar before serving.