To make the dough:
Whisk together the flour and baking powder.
Working with a stand mixer fitted with the paddle attachment, or in a large bowl with a hand mixer, beat the sugar, oil, egg, juice, vanilla and salt on medium speed for about 2 minutes, until the mixture is smooth and shiny. Turn the mixer off, add the flour all at once and pulse to incorporate, stopping to scrape down the sides and bottom of the bowl as needed. What you’ll have will look more like a batter than a dough, but that’s fine. Wrap it in plastic and refrigerate it for 1 hour, the time it takes for it to remind itself that it’s a dough.
Remove the dough from the refrigerator, divide it in half and form each half into a disk. Keep one piece in the refrigerator while you work on the other. Generously flour a piece of parchment paper, place the dough on it, flour the dough, cover with another piece of parchment and roll it out to a thickness of 1/8 inch — not thinner and, if you can manage it, not thicker either. Roll on both sides of the dough and peel away the papers a few times so that you don’t roll the paper into the dough. Flour the dough, if needed — this dough is soft and sticky. Roll out the second piece of dough, stack one piece, still between the paper, on top of the other on a baking sheet and freeze for at least 2 hours.
To make the filling: While the dough is resting, put the fruit and juice in a medium saucepan and bring to a boil. Lower the heat, cover and simmer slowly, stirring occasionally, until the fruit is soft and the liquid has been absorbed, about 12 minutes. If the liquid disappears before the fruit is soft, add a little more (the additional liquid can be water). Add the honey, jam and nuts and stir over low heat for about 3 minutes. Scrape the chunky jam into a bowl and cool to room temperature.
To fill and bake the hamantaschen: Center a rack in the oven and preheat it to 375 degrees F. Line two baking sheets with parchment paper or silicone baking mats. Have a 2¾- to 3-inch-diameter round cookie cutter at hand.
Working with one piece of dough at a time (keep the other frozen — even after chilling, the dough will be very soft), peel away both pieces of paper and put the dough back on one piece. Cut out as many circles as you can. I find it best to transfer the rounds of dough to the lined baking sheet as they’re cut. (If you find the dough too soft to work with, just pop the sheet of circles back into the freezer for about 15 minutes.) Place a heaping teaspoonful of filling in the center of each cookie. Lift two sides of the cookie and pinch together the point where they meet. Lift the remaining side and pinch together the points where it meets the other two sides of the dough; you’ll have a triangle with a mound of filling peeking above the dough.
Bake the hamantaschen for 15 to 17 minutes, rotating the sheet after 8 minutes, or until the cookies are deep brown around the edges and the pinched-together points and paler in the center. (While this batch is baking, you can form the second batch — stow them in the fridge until the oven is free.) Leave the cookies on the baking sheet for a couple of minutes, then carefully transfer them to a rack to rest until they are only just warm or at room temperature.
Bake the second batch on cooled baking sheets.
The dough can be made, rolled out, wrapped airtight and stored in the freezer for up to 2 months. You can make the filling up to 1 week ahead and keep it in a tightly sealed container in the refrigerator. However, once baked, the hamantaschen are best eaten soon after they come from the oven. If you want to keep them overnight, wrap them in plastic. They will be softer the next day, but still very good.