Place prunes and apricots in a large pot, cover with water, and bring to a rolling boil over high heat. Cover and cook uncovered, stirring occasionally, until fruit is tender, about 15 minutes. Drain fruit in a colander and mash together in a medium bowl using a fork. Set aside to cool while you make dough.

Preheat the oven to 350 degrees F (175 degrees C). Line baking sheets with parchment paper.

Whisk together 1 cup sugar, eggs, oil, lemon zest, and orange zest in a large bowl until well combined. Sift together flour and baking powder in another large bowl. Stir egg mixture into flour mixture, kneading with hands until dough comes together.

Transfer dough to a lightly floured surface and roll out dough to about 1/4-inch thickness. Cut circles out using a cookie cutter or the rim of a drinking glass.

Mix together prune-apricot mixture, walnuts, remaining 1/3 cup sugar, and lemon juice in another large bowl until filling is well combined. Place 1 tablespoon filling in the center of each dough circle. Pinch the edges firmly together to create a triangle, leaving the center open to expose filling. Repeat with remaining dough circles. Place hamantaschen on the prepared baking sheets.