Make the cupcakes: Preheat the oven to 350 degrees F. Line a 12-cup muffin pan with liners. Whisk the flour, baking powder, baking soda, salt, cinnamon and nutmeg in a medium bowl. Whisk the eggs, granulated sugar, brown sugar, melted butter, walnut oil and vanilla in a large bowl until smooth. Whisk in the flour mixture, then fold in the grated carrots, chopped walnuts and candied ginger with a rubber spatula.
Divide the batter among the prepared muffin cups, filling each two-thirds of the way. Bake until the tops of the cupcakes spring back when gently pressed, about 20 minutes. Let cool 5 minutes in the pan; remove to a rack to cool completely.
Meanwhile, make the candied carrots and walnuts: Coat a baking sheet with cooking spray. Peel 12 long ribbons from the carrot using a vegetable peeler. Combine the sugar and water in a small saucepan and bring to a boil. Reduce the heat to a simmer, add the carrot ribbons and cook until translucent, 25 to 30 minutes. Carefully remove with tongs and arrange each ribbon in a small bundle on the baking sheet. Add the walnuts to the saucepan and simmer until glossy, 5 minutes. Remove with a slotted spoon and place on the baking sheet; let the candied carrots and walnuts cool.
Make the frosting: Beat the butter, cream cheese, vanilla and salt in a large bowl with a mixer on medium-high speed until smooth. Beat in the confectioners¿ sugar on low speed; increase the speed to medium high and beat until fluffy, 3 minutes. Transfer to a piping bag with a large round tip; pipe onto the cupcakes. Top with the candied carrots and walnuts.