Carrot-Walnut Cupcakes

Name: Carrot-Walnut Cupcakes
Cook time: 1 h
Total time: 1 h
Cuisine: American
Ingredients: 1 cup all-purpose flour; 3/4 teaspoon baking powder; 1/2 teaspoon baking soda; 3/4 teaspoon salt; 1 teaspoon ground cinnamon; 1/4 teaspoon ground nutmeg; 2 large eggs; 1/2 cup granulated sugar; 1/3 cup packed light brown sugar; 1/3 cup (51/3 tablespoons) unsalted butter, melted; 1/3 cup walnut oil or vegetable oil; 1 teaspoon pure vanilla extract; 1 3/4 cups finely grated carrots (about 2 medium); 1/2 cup walnuts, roughly chopped; 1 tablespoon finely chopped candied ginger; Cooking spray; 1 large carrot, peeled; 3/4 cup granulated sugar; 3/4 cup water; 12 walnuts; 6 tablespoons unsalted butter, at room temperature; 1 8-ounce package cream cheese, at room temperature; 1/8 teaspoon salt; 3 1/2 cups confectioners¿ sugar, sifted