In the bowl of a stand mixer with the paddle attachment, beat the eggs, sugar, melted butter, vanilla extract, and water. Mix for 2 minutes on medium speed.
Next, add the lemon zest, juice, & lemon pulp, and mix on low speed for 1 minute.
Add the flour, baking soda, baking powder, and salt. Mix until combined.
Remove the dough from the bowl and wrap tightly in plastic wrap. Refrigerate at least 1 hour.
Preheat your oven to 375F. Prepare 2 Cookie sheets with parchment paper and place aside.
After chilling, unwrap the dough and cut into thirds. Place 1/3 of the dough on a well floured surface and begin to roll the dough out with a floured rolling pin. Continue to roll into a loose circle, about 1/8 of an inch thick.
Take a 3 inch diameter round cookie cutter (or drinking glass, as I did here), and begin to cut out circles. Be mindful of where you choose to use your cutter to minimize waste.
Once all the circles have been cut, reserve the scraps for one more roll out. You can get another couple of cookies out of the scraps, but I wouldn't recommend a 3rd roll out with the same dough because the texture gets rather tough.
Place a heaping teaspoon of your favorite filling into the middle of each circle.
Pinch 3 corners (very tightly) with slightly water dampened fingers, to create the 3 sides.
Place the finished cookies carefully onto your prepared cookie sheets.
Repeat the rolling/cutting/filling/pinching process with the rest of the dough.
Bake the cookies for 12-14 minutes, or until the edges are slightly crisped and the bottoms are lightly browned.
Once baked, transfer to a wire rack to cool completely.
Baked Hamentashen can be stored in a covered tin or tea cloth draped plate for about 3 days, before going stale. They also freeze well in a tightly covered container, for up to a month.