Add the flour, sugar, baking powder, and salt to the bowl of a food processor. Pulse a few times until well blended.
Add the butter. Pulse a few times to cut the butter into the flour, creating sandy-textured mixture.
Add the egg, vanilla, and 1 teaspoon of lemon juice. Pulse until it forms a ball of dough. If dough doesn’t easily come together in 1 to 2 minutes, add the remaining teaspoon of lemon juice.
Scrape the dough onto a lightly floured board. Gently knead a few times until smooth, and no longer sticky. Wrap in plastic film, and chill until firm, but still soft enough to roll out. Dough may be prepared a day in advance. Let it sit at room temperature 15 to 20 minutes to allow it to soften up enough to roll out.
When ready to shape and bake, preheat the oven to 375ºF (190ºC). Line to sheet pans with parchment paper. Keep a small bowl of water nearby.
Divide the dough into quarters. Take one piece, and place on a lightly floured counter. Sprinkle some flour over the dough, and roll out ¼-inch thick. Using a 2 ½-inch cutter, press out circles. You may reroll the scraps to use it all up.
Dip your fingertip in water, and rub around the edges of the dough circles. Drop a ½ teaspoon of Poppy Seed Filling into the center of each piece of dough. To fold the hamantaschen, use your pointer finger and thumb on both hands to gather the ends of the dough, and pinch them closed (you should have two sealed corners at the bottom). Pinch the top of the circle closed to form a triangle. Repeat with the remaining circles, and repeat the entire process again with one more piece of the dough. Arrange shaped and filled cookies on one of the prepared pans. Bake until edges and bottoms are golden, 15 to 17 minutes.
Meanwhile, repeat shaping and filling more cookies with the remaining dough while the first batch bakes. Let cookies cool completely on a wire rack before packing in a tin. Best eaten in a few days.