Preheat the oven to 350 degrees F (175 degrees C). Grease the bottoms only of 12 muffin cups, or line with baking cups. Sift together flour, cinnamon, baking soda, baking powder, and salt; set aside.

Beat eggs and sugar until frothy and lightened in color. Stir in oil, vanilla extract, and grated carrot; fold in flour mixture. Pour batter into prepared muffin tins.

Bake in the preheated oven until a toothpick inserted in the center comes out clean, 15 to 20 minutes. Remove to cool on wire racks.

Meanwhile, prepare frosting by beating cream cheese, butter, vanilla, and confectioners' sugar together until fluffy. Frost the cooled muffins and chill until ready to serve.