Preheat the oven to 350°F (175°C). Grease a 12-cup muffin pan or line with cupcake liners and set aside.
In a large bowl, beat the eggs briefly until combined. Stir in the grated carrots and the oil until evenly mixed.
Add the flour, granulated sugar, cinnamon, baking soda, baking powder, and salt to the carrot mixture. Stir gently until just combined; do not overmix.
Divide the batter evenly among the prepared muffin cups, filling each about two-thirds full.
Bake for 18–20 minutes, or until the cupcakes are golden and a toothpick inserted into the center comes out clean. Remove from oven and cool in the pan for about 10 minutes, then transfer to a wire rack to cool completely.
To make the frosting, beat the room-temperature butter and cream cheese together until smooth and fluffy. Add the vanilla, then gradually beat in the powdered sugar. Add the heavy cream 1 tablespoon at a time if needed to reach a spreadable consistency.
Once cupcakes are completely cool, frost them with the cream cheese frosting using a piping bag, a zip-top bag with the corner snipped, or a spoon.