Preheat the oven to 350 degrees and line 2 sheet pans with parchment paper.
In a medium bowl, using a stand mixer or hand mixer (whichever can handle cold butter or let the butter soften some before starting), blend butter and sugar. Add the egg and vanilla. Mixing well to combine.
Combine the flour and salt in a bowl, then gradually add the mixture a little at a time with the mixer on a low speed, so the flour doesn’t fly all over. You can do this by hand if you prefer, just be sure it is well combined.
At this point, you can start rolling and cutting out the dough, but if it seems too sticky, dust with powdered sugar and try to roll it out. If it's still too sticky, wrap it in plastic wrap and refrigerate it for about 30 minutes. If it seems a bit dry and is cracking when you roll it out, knead about a teaspoon of milk or heavy cream into the dough by hand. Continue adding milk if needed until it becomes pliable without cracking.
On a clean surface, sprinkle powdered sugar. Form the dough into a disk and sprinkle powdered sugar on top to prevent the rolling pin from sticking.
Roll the dough out to about ⅛ inch thick and cut 2 ½” circles with a cookie cutter.
Using a spatula, carefully lift and transfer the round pieces of dough to the lined baking sheet. These cookies don’t spread, so you can put them about ½ inch apart. Continue pulling the scraps together into a ball and rolling again until all of the dough has been cut out into cookies.
Spoon about a teaspoon of filling into the center of each circle of dough. If it seems a little dry, dip your finger into water or milk and make a ring on the outside edge of the dough. Gently pull three sides of the circle together to form a triangle and press the dough together at each seam to seal the shape. Another dip of a finger into the water or milk will help to seal the dough as well if needed.
Bake for approximately 8-10 minutes. Remove before the edges begin to brown. Let them cool completely on the baking sheet before transferring them to a serving tray. They will crumble if you try to remove them before they are cool.