In a large bowl, beat the butter and sugar together on medium-high speed for 1-2 minutes, until light and fluffy.: The kitchen should begin to smell faintly sweet and buttery as the unsalted butter and granulated sugar come together, and you will see the mixture lighten in color and texture, becoming airy. This step creates tiny pockets of air that help produce a tender crumb, and you may hear the mixer humming steadily as it aerates the fats. If the mixture stays dense, it usually means the butter was too cold, so let it warm up a bit before retrying. Avoid overbeating, which can make the butter too soft and cause spreading later.
Add the egg, lemon zest, and vanilla and mix until fully combined. Scrape down the sides and mix again for 10-20 seconds.: A bright citrus aroma from the lemon zest will pop forward, and the batter will gain shine as the egg incorporates, binding the mixture. Scraping the bowl ensures no pockets of dry sugar or butter remain, giving an even texture. If the mixture looks curdled after adding the egg , it usually smooths with a brief additional mix; if not, the butter was likely too warm. Avoid continuing to mix for too long, which can develop gluten once the flour is added.
Add the flour and salt and mix until just barely combined in a crumbly dough. Use your hands to press the dough together. If the dough is too dry, add water one tablespoon at a time to help it come together. Be careful not to add too much water, it should only be enough to bring the dough together, not enough to make it sticky. Form the dough into a disk and wrap it with plastic wrap. Chill for 2 hours. While the dough is chilling prepare the poppy seed filling.: When you add the all purpose flour and salt , the texture will shift from glossy to a pebbly, crumbly mass. Pressing it together by hand warms the fats slightly, helping the dough cohere without becoming greasy. The chilling time firms the butter so the cookies hold their shape when baked, and resting also lets flavors meld. If you add too much water , the dough becomes sticky and will be difficult to roll, so add water only sparingly. A common mistake is skipping the chill, which can cause the cookies to spread and lose their triangular shape during baking.
Once the dough is chilled, preheat the oven to 350°. On a lightly floured surface, roll the dough out to 1/8 inch thick. You may need to let the dough sit for a few minutes so it is a little easier to roll out without it cracking or crumbling. If it does crack or crumble, use your fingers to pinch the dough back together.: As the oven warms to 350 degrees , the chilled dough becomes easier to roll. The surface should feel cool and slightly firm under your hands, and the dough should roll smoothly to a thin, even sheet about the thickness of a playing card. If it cracks at the edges, a brief rest at room temperature softens it enough to mend without overworking. Overworking creates gluten, making the cookies tough, so be gentle and patch rather than re-rolling repeatedly.
Use a 3-inch round cooking cutter to cut out rounds of dough. Carefully transfer the rounds to parchment-lined baking sheets leaving 1 inch between them. Place 1 teaspoon of poppy seed filling or jam in the center of the dough. Fold up the side of the dough to form a triangle around the filling, pinching the corners together to prevent leaking while baking. You can rub a little water or beaten egg on the corners before pinching it together to help act as glue to keep it from opening. Gather the remaining dough and repeat, though the more the dough is rolled the tougher the cookies will be.: The round shapes should be uniform so the triangles bake evenly, and when you place a teaspoon of filling in the center you will see a small mound of color or texture. Folding the sides into a triangle creates a snug pocket; the sound of the dough pinching together is quiet and satisfying. Using a dab of water as glue helps the corners seal, but be sparing since too much moisture can make the edges soggy. Repeated rolling toughens the dough, so try to cut as many rounds as possible from a single roll and reserve scraps for a gentle re roll.
Bake for 12-15 minutes, or until the cookies are just starting to turn golden brown around the corners. Let them cool on the baking sheet for 10 minutes before transferring to a cooling rack.: As the cookies bake you will notice a gentle deepening of color at the corners and a warm, sweet aroma filling the kitchen. The edges should be only slightly golden, not deeply browned, keeping the interior tender. Letting them rest on the warm sheet off the heat helps the cookies set so they do not collapse when moved; they will continue to firm as they cool. Removing them too soon risks breaking the bases, but leaving them too long on a hot pan can overbrown the bottoms.
Use a spice or coffee grinder to grind the poppy seeds into a powder. Transfer poppy seed powder to a medium bowl and add the sweetened condensed milk. Stir until fully combined. Cover and keep in the refrigerator until ready to use.: Grinding the poppy seeds releases oils and intensifies their nutty aroma, which is much more pronounced than when they are left whole. When combined with the sweetened condensed milk , the mixture becomes glossy and thick, perfect for spooning into the cookie centers. Refrigeration firms the filling slightly and melds flavors. A common pitfall is under grinding, which leaves a gritty texture, so process until a fine paste forms but avoid turning it into a paste that is too thin.