Unwrap the dough disk and place it on a lightly floured work surface. It will be very firm after chilling.
Roll the dough out to ¼-inch thick. It might be stiff at first and you may need to pound it with the rolling pin to soften enough to roll out. If large cracks form around the dough edges, just repair them with your fingers as you roll. Use a bench scraper under the dough occasionally to keep the dough from sticking to the work surface, and also lightly flour the rolling pin and surface occasionally to prevent sticking.
When the dough reaches ¼-inch thickness, you can either continue rolling out to be closer to about ⅛-inch thick (very thin) or leave it closer to ¼-inch thick. See the Recipe Notes for the pros and cons of dough thickness.
If the dough has gotten too soft to cut, transfer it to the half sheet baking pan and chill it for 5 minutes to firm up. When it's ready, cut out as many circles from the dough as you can using a 3-inch round cookie cutter or the 3-inch rim of a glass, but no smaller. Gather the scraps, roll out the dough and cut more circles. Cover the circles with a lightly damp towel while they wait to be filled.
Place a teaspoon of filling into the center of each circle. Too much filling will cause the Hamantaschen to open and the filling to spill out while baking.
Shape the Hamantaschen: Fold the bottom of the circle upwards towards the center, making a flap covering the bottom third of the circle. Take the left side and fold it towards the center, overlapping the side corner of the bottom flap to create a tip at the bottom left corner of the circle. A small bit of filling will still be visible in the center.
Now take the right side of the circle and fold it towards the center to create a third flap and completing the triangle. Tuck the bottom of this flap under the bottom right corner of the triangle while letting the top of the flap overlap the top corner of the triangle. Each side of your triangle should have a corner that folds over and a corner that folds under, creating a pinwheel look that helps to keep the cookies from opening while they bake. It's also pretty!
Press to seal the corners triangle gently but firmly to secure the shape. If any cracks have formed at the places where the dough is creased, use slightly wet fingers to smooth them out.
When all of your Hamantaschen have been filled, place them on the half sheet baking pan evenly spaced and chill them for about 5 to 10 minutes before baking. Like with all butter cookies, this will help the Hamantaschen keep their shape while baking.
Bake the cookies for 10 to 25 minutes, checking after 10 minutes. They should be lightly golden brown, and the thickeness will affect the time it takes to bake.
Cool the Hamantaschen on a wire rack, then store them in an airtight container. Enjoy, making as much noise as possible!
The dough should be smooth and slightly tacky, but not sticky. If the dough is too dry, add water slowly in 1 teaspoon increments and using your hands to knead the liquid into the dough. If the dough seems too wet, knead in 1 teaspoon of flour, again kneading gently until the dough reaches the right texture. The thinner you roll the dough closer to ⅛-inch thick, the more delicate and crisp the cookies will turn out, but too thin and the dough will tear as you shape them into triangles. Keeping the dough closer to ¼-inch thick will yield a doughy, less delicate texture to your cookies, but can crack as you shape them. Bottom line, make sure that the dough is thick enough to hold the filling without tearing or cracking during shaping.The butter cookie dough can be wrapped well in plastic and stored in an airtight plastic bag in the freezer for up to 3 months. Defrost overnight in the refrigerator and proceed with assembly. The finished cookies themselves are best the same day, but can be frozen for a few weeks.