Preheat oven to 350°F with rack in middle. Line muffin tins with cupcake liners.
In a large bowl whisk together flour, baking powder, baking soda, salt, ginger, cinnamon and nutmeg.
In separate bowl whisk together eggs, oil, brown sugar, grated carrots, chopped walnuts and vanilla. Stir in flour mixture until just combined- don't over-mix.
Divide batter evenly among muffin tins and bake until golden and a wooden pick inserted into center of a cupcake comes out clean, approz 20 to 25 minutes.
Cool in pan approx 10 minutes. Remove cupcakes from pan and cool completely before frosting.
Mix the cream cheese and butter together in a bowl. Slowly add the sifted confectioners' sugar- starting with 2 cups of sifted confectioner's sugar and adding more in increments until desired consistancy is reached (depending on climate this may be another 1 to 2 cups). Stir in the vanilla and lemon juice. Frost cupcakes.