Shred carrots:

Remove carrot skins and shred them. If using a coarse grater, mince the shreds finely with a knife.

Mix dry ingredients:

In a bowl, beat eggs with oil until blended. In a separate bowl, mix flour, baking powder, brown sugar, granulated sugar, vanilla sugar, and cinnamon.

Combine wet and dry:

Add the wet mixture to the dry ingredients and fold gently until barely combined; avoid overworking. Then incorporate the grated carrots with slow folding to ensure even distribution.

Fill muffin cups:

Portion the batter into 12 paper-lined muffin cups, filling each to three-quarters full. Bake at 350°F (175°C) for 18-20 minutes, until a skewer inserted in the center emerges clean.

Cool muffins:

Allow muffins to cool fully on a wire rack.

Make cream cheese frosting:

Using an electric mixer, cream together the softened butter and cream cheese. Slowly incorporate confectioners' sugar, beating until the mixture is airy and smooth. For flavor, mix in lemon juice; adjust consistency with milk if desired. If too runny, add extra confectioners' sugar; if too stiff, add milk a little at a time until the frosting is pipeable.

Pipe frosting:

Transfer frosting to a piping bag fitted with a tip, then pipe onto the fully cooled cupcakes.