Preheat the oven to 350°F (180°C).
Butter or spray an 8-inch pan. You can line it with a piece of parchment paper as I do. It’s easier to remove after it’s baked. Or you can dust with fine breadcrumbs or flour.
Beat ½ cup + 1 tablespoon unsalted butter with 1 cup light brown sugar in a large bowl at medium-high speed for 2-3 minutes, until creamy, smooth, and the sugar has partially dissolved.
Add 2 eggs, one at a time, and beat well after each addition but no more than half a minute or so.
Sift over this mixture the dry ingredients (¾ cup all-purpose or cake flour, ¾ cup superfine whole wheat flour, 2 teaspoons baking powder, ½ teaspoon ground cinnamon, ½ teaspoon ground ginger, pinch of ground allspice, pinch of ground nutmeg, ½ teaspoon freshly grated ginger and ¼ teaspoon salt. You can sift them beforehand in a bowl or have them measured and sift them directly over the batter.
Beat at low or medium speed *just* until combined. Don't overmix at this point.
Add 2 cups grated or shredded carrots (it’s best to grate them at the last minute, so they keep their moisture but don’t release liquid) and mix well. I do this with a rubber spatula.
Add ¼ cup milk and ½ teaspoon vanilla extract and mix until well incorporated.
Pour batter into the prepared pan, smooth top and bake for about 35-40 minutes, until springy if you lightly touch the center and a cake tester or toothpick inserted in the middle comes out clean.
Let cake cool on a wire rack until warm enough to handle and carefully remove from the pan onto the serving plate.