"After trying a wide range of different approaches, I finally hit the right combination of ingredients for a lush, moist, and intensely tasteful carrot cake batter. Do yourself a favor and try it ASAP! Optional: Decorate with flakes of hazelnut and slices of lemon peel."
Mix 1/2 cup plus 2 tablespoons oil, eggs, brown sugar, and grated carrots together, in that order, in a bowl.
Mix flour, cinnamon, baking powder, vanilla sugar, and salt together in a separate bowl. Add flour mixture to oil mixture and stir to combine. Fill 12 cups of a muffin tin with the batter.
Bake on the middle rack of the preheated oven until a toothpick inserted in the center comes out clean, 12 to 15 minutes; do not overbake. Cool completely before frosting, about 45 minutes.
Mix confectioners' sugar, cream cheese, butter, lemon juice, and lemon zest together in a bowl; stir until the icing reaches a smooth, thick, and creamy texture. Cover icing with plastic wrap and refrigerate until set, at least 30 minutes, before topping cupcakes.