Position a rack in the middle of the oven and heat the oven to 350 degrees F. Butter an 8-inch square baking pan.

In a medium bowl, whisk 1 cup (4-1/2 ounces) of the flour with 1/4 cup of the confectioners' sugar, the lemon zest, 1/2 teaspoons of the ginger, and a pinch of salt. Cut in the butter with a pastry blender or two table knives until the mixture resembles small peas. Knead the dough in the bowl just until it begins to come together. Transfer the dough to the baking pan and, with floured hands, press it evenly over the bottom. Bake until very light golden brown, about 20 minutes. Let cool on a rack while you make the filling.

In a small bowl, whisk 1-1/2 cups of the confectioners' sugar, the remaining 2 tablespoons (1/2 ounce) flour, the baking powder, and a pinch of salt. In a medium bowl, beat the eggs with an electric mixer on high speed until tripled in volume, pale yellow, and very light and fluffy, 3 to 5 minutes (the eggs will hold soft peaks very briefly). Reduce the speed to low, add the sugar and flour mixture, and beat just until blended, scraping the bowl as needed. Add the lemon juice and beat just until blended. Pour the lemon mixture over the warm crust.

Bake until the filling is just set in the center, is golden brown on top, and doesn't jiggle when the pan is nudged, 18 to 20 minutes. Let cool completely in the pan on a rack. Just before serving, stir the remaining 1 teaspoon confectioner's sugar with a pinch of ginger in a small bowl. Transfer to a small seive and sift over the lemon filling. Using a bench scraper or metal spatula, cut into 2-inch squares, slicing straight down (rather than dragging). Store in an airtight contianer.