Heat oven to 350 degrees. Spread pecans in a single layer on an ungreased baking pan. Transfer to oven, and roast until toasted and brown, about 8 minutes. Give the pan a shake midway through roasting to make sure they roast evenly. Remove pan from oven, and let stand until completely cool. Finely chop pecans, and set aside. Line muffin tin with cupcake liners. Set aside.
Using the smallest holes (less than 1/4-inch in diameter) of a box grater, grate carrots to yield 1 3/4 cups. Place grated carrots, eggs, buttermilk, vanilla, sugar, vegetable oil, and ginger in a large bowl; whisk until well combined.
In a medium bowl, whisk together flour, baking powder, baking soda, salt, and cinnamon. Using a rubber spatula, fold the dry ingredients into the carrot mixture until just combined. Fold in toasted pecans.
Using a large metal spoon, fill each cupcake to the top of the paper liner. Bake until a cake tester comes out clean, rotating the pan halfway through, 20 to 25 minutes.
Let cool in pan for 10 minutes, then remove to cooling rack to cool completely. Spread about 1/ 4 cup frosting on each cupcake and serve. Store in an airtight container up to two days.