To prep the preserved lemon peel, cut 2 preserved lemons into quarters. Pull the peel away from the flesh, and finely chop the peel. Discard the inner flesh.
In the bowl of a stand mixer, combine the preserved lemon peel, butter, and 1 cup sugar. Beat on medium speed until light and fluffy, scraping down the sides and bottom of the bowl halfway through, 2-3 minutes. Turn the mixer off, add the eggs and vanilla, and mix until pale yellow and very light, another 2-3 minutes. Turn the mixer off, add the flour and baking powder, and mix on low speed just until a soft dough forms, 1-2 minutes. Tightly cover the bowl with plastic and chill in the refrigerator for at least 2 hours. Chilling overnight or a few days is fine.
Place the remaining 1/3 cup sugar in a shallow wide bowl, and the powdered sugar in a second wide shallow bowl. Scoop the cookie dough into about 2-tablespoon-sized portions. Roll into balls. Roll each ball in granulated sugar, then in powdered sugar. Then roll in the palms of your hands to reshape and smooth them. Place on a small board or container and freeze at least for an hour or overnight.
When ready to bake, heat the oven to 350F and set the oven racks at the upper-middle and lower-middle positions. Line 2 baking sheets with parchment paper. When the oven is hot, then remove the frozen dough balls from the freezer and arrange on baking sheets, 3 inches apart. Bake both sheets at the same time for about 13-15 minutes, rotating when they are halfway through. Bake until the cookies have puffed and cracks no longer look wet.
Let cool completely on baking sheets or a cooling rack. Cookies will keep in an air-tight container at room temp for a week.