In a large bowl, whisk together the flour, sugar, baking powder, and salt. Add the butter and rub it into the flour mixture with your fingertips until it resembles coarse crumbs.Add the eggs, lemon zest, and vanilla. Mix until a dough forms, then knead briefly until smooth and supple, about 3–4 minutes. The dough should be firm but not sticky. If it feels dry, add a few drops of water; if sticky, dust lightly with flour.Form into a disk, wrap tightly, and let rest at room temperature for 30 minutes.
Roll the Dough
Divide the dough into 4 pieces, keeping the portions you’re not working with covered.Using a pasta machine:Flatten one piece slightly and run it through the widest setting.Continue passing it through the machine, reducing the thickness one notch at a time, until you reach setting 6 or 7 (very thin, almost translucent).Using a rolling pin:Roll the dough out on a lightly floured surface until it is paper-thin. You should almost be able to see through it.
Cut the Cookies
Using a fluted pastry wheel, cut the dough into rectangles or diamond shapes, about 2 x 4 inches. Cut a small slit in the center of each piece if desired (traditional but optional).Place the cut dough on a lightly floured surface and repeat with remaining dough.
Fry
Heat about 2 inches of oil in a deep skillet or Dutch oven to 350°F (175°C).Fry the cookies in batches, turning once, until puffed and lightly golden, about 30–45 seconds per side. Do not overcrowd the pan.Transfer to a paper towel–lined tray to drain.
Finish
Once completely cool, dust generously with powdered sugar.