Pulse the flour, granulated sugar, 1/2 teaspoon salt and the baking powder in a food processor. Add the shortening and 4 tablespoons butter; pulse until pea-size pieces form. Add the lemon juice and 5 tablespoons ice water; pulse to combine. Gradually pulse in 1 more tablespoon water until the dough clumps together. Turn out onto a sheet of plastic wrap; shape into a disk. Wrap and refrigerate until firm, at least 1 hour.
Combine the dried fruit, the remaining 2 tablespoons butter and 1/3 cup water in a microwave-safe bowl. Cover and microwave until softened, about 3 minutes. Let stand 10 minutes, then transfer the fruit and liquid to a food processor. Add the honey, cinnamon and a pinch of salt; pulse to make a paste.
Position racks in the upper and lower thirds of the oven; preheat to 375 degrees F. Line 2 baking sheets with parchment. Roll out the dough on a floured surface until 1/8 inch thick. Cut out circles with a 3-inch cookie cutter. Reroll the scraps and cut out more circles.
Spoon 1 teaspoon filling onto each circle; brush the edges with the beaten egg. Fold the dough over the filling and press the edges to seal; transfer to the baking sheets. Brush with more beaten egg and sprinkle with coarse sugar; cut slits in the tops. Chill until firm, 15 minutes.
Bake, switching the pans halfway through, until the cookies are golden brown, 25 minutes. Let cool completely on the baking sheets.