Combine chopped dates, sugar, and water in a 2-quart saucepan over medium heat. Bring to a boil, then reduce to low and simmer for 10–15 minutes, stirring frequently, until thickened to a paste-like consistency.
Remove from heat and stir in the lemon juice and a pinch of salt. Transfer to a bowl and cool completely to room temperature before using — the filling will thicken further as it cools.
Beat softened butter and granulated sugar in a stand mixer on medium speed for 3–4 minutes until light, fluffy, and pale. Add eggs one at a time, beating well after each, then mix in vanilla extract.
Whisk flour, baking powder, and salt in a separate bowl, then add to the butter mixture in three additions, mixing on low just until combined after each. Do not overmix.
Divide dough into two disks, wrap tightly in plastic wrap, and refrigerate for at least 1 hour or up to 24 hours. If chilled more than 2 hours, let sit at room temperature for 10 minutes before rolling.
Preheat oven to 375°F (190°C) and line baking sheets with parchment paper. On a floured surface, roll one dough disk to 1/8-inch thickness, then cut out circles with a 2.5-inch round cutter.
Transfer half the circles (bottoms) to the baking sheet. Use a 1-inch cutter or thimble to cut a hole in the center of the remaining circles (tops), then set those aside on your work surface.
Spoon 1 teaspoon of cooled date filling onto each bottom cookie, spreading slightly and leaving a 1/4-inch border. Top with a cut-out round and press edges together firmly with fingertips or a floured fork to seal completely.
Bake on the center rack for 10–12 minutes, until the edges are just beginning to turn light golden. The tops should remain pale — watch carefully during the last few minutes.
Cool cookies on the baking sheet for 2–3 minutes, then transfer to a wire rack to cool completely (about 30 minutes). Repeat rolling, filling, and baking with the second dough disk and any scraps.