Heat oven to 400°F and line a baking sheet. In a large bowl, whisk flour, both sugars, baking powder, salt, cinnamon, nutmeg, and ginger.

Cut cold butter into the flour mixture with a pastry cutter until the largest pieces are pea-sized.

Stir in the grated carrot and walnuts. In a small bowl, whisk sour cream, egg, and 1 teaspoon vanilla. Fold into the flour mixture just until no dry streaks remain.

Turn dough onto a floured surface, pat into an 8-inch circle about 1 inch thick, and cut into 8 wedges. Arrange on the sheet and bake 18-20 minutes, until deep golden and a toothpick comes out clean. Cool on a rack.

Beat cream cheese, powdered sugar, 2 tablespoons milk, and ½ teaspoon vanilla until smooth, adding more milk if needed. Drizzle over the fully cooled scones.