Rest butter and egg on counter til at room temperature.
Slice butter into large chunks and place in a large mixing bowl. Add the sugar and mix until light and fluffy (we recommend using an electric mixer). Add the egg, vanilla and lemon zest to the bowl and beat again until well mixed and creamy. Add flour and salt and mix till you have a crumbly dough (use low speed if using an electric mixer). Start kneading the dough till a smooth dough ball forms, being careful to not overwork it. If needed, add water a teaspoon at a time. You'll want the end result to be a smooth look with a slightly tacky feel where the dough can still be rolled out.
Wrap the dough ball in plastic wrap and press it into a flat-ish disk by hand. Refrigerate for 3 hours or overnight.
When the dough has been chilled, prepare your filling before assembling the cookies. Place pitted prunes and apricots into a large pot of water and bring to a boil over high heat. Cook the fruit uncovered (stirring occasionally) for about 15 minutes or until the fruit is tender. Drain the fruit then mash in a large bowl. Add lemon juice, chopped walnuts, and 1/3 cup sugar into the bowl and mix well.
Take out the dough and use a rolling pin to roll the dough out to about 1/4 inch thick. A heavy rolling pin and a good pounding helps with the cold dough. When it reaches 1/4 inch, use a pastry scraper to peel the dough off the counter, reflour the surface, and flip the dough. Roll it out more until it's 1/8 inch thick. The thinner you get the dough the more light and crisp your cookies will be.
Use a 3 inch circular cookie cutter to cut the cookies out of the dough. You should end up with about 35 cookies after using all the dough scraps.
Preheat the oven to 350 degrees F.
Dampen a towel and place it over the dough circles while you load them to prevent drying and cracking. Take one circle at a time and place a teaspoon of filling in the center of each circle.
Fold the hamantaschen in three steps. First, take the left side of the circle and fold it towards the center to make a flap that covers the left third of the circle, keeping in mind this is the first side of an even triangle. Then take the right side of the circle and fold it towards the center, overlapping the upper part of the left side flap to create a triangular tip at the top of the circle. A small triangle of filling should still be visible in the center. Grasp the bottom part of the circle and fold it upward to create a third flap and complete the triangle. When you fold this flap up, be sure to tuck the left side of this new flap underneath the left side of the triangle, while letting the right side of this new flap overlap the right side of the triangle. This way, each side of your triangle has a corner that folds over and a corner that folds under, creating a "pinwheel" effect that is not only pretty, but will help to keep the cookies from opening while they bake.
Place the cookies on a lightly greased cookie sheet, evenly spaced at least an inch apart. Place the cookie sheet in the oven and bake for 20-25 minutes, or until they're lightly golden.
Cool the cookies on a wire rack and store in sealed containers to retain moisture.