Put whole lemons in a large pot over high heat, cover with cold water, and bring to a boil. Lower heat to medium and simmer until soft, about 30 minutes. Drain; transfer to a bowl of ice water. When lemons are cool, cut into quarters and gently remove seeds, retaining as much juice as possible. Whirl lemons in a food processor until they become a smooth, thick puree.
Meanwhile, preheat oven to 350°F The original recipe said to butter and flour a 9-inch pan with a removable rim. When I made it, there was far too much batter for my 9-in. tart pan. I used a 12-inch pan and still had a little extra.
In a food processor, pulse almonds until nuts resemble coarse cornmeal. Put in a large bowl and add 1/2 cup flour, baking powder, and salt. Stir to combine.
In another large bowl, beat together egg yolks and sugar until thick and pale yellow. Stir in lemon puree and almond extract. Add almond mixture and stir to combine. Stir in candied ginger. Set aside.
In another large bowl, beat egg whites to form firm peaks. Gently fold egg whites into batter. Spread batter in prepared pan.
Bake until edges of cake begin to pull away from the sides of the pan, about 1 hour. Cool on a rack for 15 minutes, then remove from pan and cool completely.
In a small bowl, combine powdered sugar and enough lemon juice to make a glaze. Spread glaze over cake. Garnish with lemon slices.