In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.

In a large bowl, beat the softened butter and 1 cup sugar together on medium speed until light and fluffy, about 2 minutes. Scrape the bowl.

Add the egg, lemon juice, lemon zest, and vanilla extract. Beat on low until combined. The mixture may look slightly curdled — that's fine.

Gradually add the dry ingredients to the wet ingredients, mixing on low just until no streaks of flour remain. Do not overmix. The dough will be soft but not sticky.

Divide the dough in half, flatten each half into a disk about 1/2-inch thick, wrap in plastic wrap, and refrigerate for at least 1 hour (or up to 2 days). Chilling firms the butter and prevents spread.

When ready to bake, preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.

In a small bowl, stir together the 1/4 cup sugar and 1 tablespoon lemon zest for the coating. Rub the zest into the sugar with your fingers until fragrant.

On a lightly floured surface, roll one chilled dough disk to 1/4-inch thickness. Use a bench scraper to lift and rotate the dough occasionally to prevent sticking.

Cut shapes with cookie cutters, or simply slice into squares with a knife. Gather scraps, re-roll once, and cut more. Discard the final scraps.

Place cookies 1 1/2 inches apart on the prepared baking sheets. Sprinkle generously with the lemon-sugar coating before baking.

Bake for 9-11 minutes, rotating the pans halfway through. The edges should be set and just barely golden, but the centers will look soft and slightly puffed. Do not overbake.

Let cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely. They will firm up as they cool.