Prepare the Dough. In a bowl, whisk together the flour, baking powder, ginger, cinnamon, nutmeg and salt. Put aside.
In a mixer, cream the butter and sugar together for 2-3 minutes. Add the egg, the egg yolk, milk, and vanilla and beat until all the liquid ingredients are fully incorporated.
With the mixer off, pour the dry ingredients into the mixer bowl over the liquid ingredients. Turn the mixer on the lowest speed and mix just until the dry ingredients are fully incorporated. (Alternatively, you can mix the dry ingredients in by hand using a rubber spatula.) Do not overmix.
Place half the dough on top of a large piece of plastic wrap. Using the plastic wrap, shape the dough into a disk, pushing down on it to thin it out. Fully cover the disk of dough in the plastic wrap and place in the refrigerator for at least two hours and for as long as 24 hours. (I usually make the dough the day before I am going to bake, so I let it sit overnight in the refrigerator.) Repeat with the remaining portion of the dough. While the dough rests in the refrigerator, make the apple filling.
Make the Apple Filling. Cover a jelly roll pan or cookie sheet with plastic wrap. Set aside.
Cut the apples into ¼ of an inch squares and place into a 2-quart saucepan.
Add the sugar, vanilla bean paste (or extract), cinnamon stick, lemon juice and water to the saucepan.
Gently mix all the ingredients together. Cover the saucepan with its lid tilted to the side so that steam can escape easily. Cook mixture on a low flame, stirring occasionally (every 5-7 minutes). The mixture is done when the apples are translucent and the water has evaporated, approximately 30-35 minutes. (If some of the apples cook to the point of applesauce, do not panic. It will taste just as good!)
Pour the cooked apple mixture onto the plastic wrap and allow it to cool. (You can also pour it into a heatproof dish to cool.) Once cool, remove the cinnamon stick and discard. If not using immediately, transfer the apple mixture to a container and refrigerate for up to seven days. (If refrigerating, you can keep the cinnamon stick in there to add some additional flavor.)
Line 2 or 3 cookie sheets with parchment paper.
Shape Hamentaschen. Take one of the dough disks from the refrigerator and roll out in between two pieces of parchment paper. Roll the dough so that it is approximately ¼ of an inch thick. If you roll it too thin, the dough will rip as you remove it from the parchment. Using a round cookie cutter (I use a 2½ inch diameter), cut as many rounds out as you can. You may need to dip your cutter into some flour to avoid it sticking to the dough. Use a metal spatula dipped in flour to release the rounds. Gather the scraps and roll that clump of dough out and cut more rounds. Keep repeating until you have used all the dough. If at any point the dough becomes too sticky or difficult to work with, cover it in plastic wrap and let it cool in the refrigerator for about 15 minutes.
Fill each round with approximately 1 teaspoon of the apple filling. Do not over fill them! Lightly flour your fingertips and form each circle into triangles by folding up the rounds and pinching them tightly in three corners. (If they are not sticking together, wet your fingertip and rub it on the inside where you are trying to create the point.) You want to make sure that the seams are fully closed so that the filling does not break through the barriers while baking. Place the shaped hamantaschen on the prepared cookie sheets.
Place the cookie sheets with the shaped hamantaschen in the freezer for at least 15 minutes. DO NOT SKIP THIS STEP.
Preheat oven to 400° F.
Repeat steps 11 through 12 with the remaining disk of dough.
Bake the frozen hamantaschen for 9-10 minutes until they are lightly brown around the bottom edges.
Cool completely on pan or a wire rack before serving.