Pulse 2 1/2 cups flour, 1/2 cup granulated sugar and the salt in a food processor until combined. Add the butter and pulse until the mixture looks crumbly. Drizzle in 1 beaten egg and pulse until the dough holds together when pinched. Turn out onto a clean surface and knead a few times until the dough comes together. Divide the dough into 2 balls and place each ball on a separate piece of parchment paper. Top each with another piece of parchment and roll out each into an 11- to 12-inch round (about 1/8 inch thick). Refrigerate until firm, about 1 hour.
Position racks in the upper and lower thirds of the oven; preheat to 375 degrees F. Line 2 baking sheets with parchment paper; set aside. Whisk the remaining 1/2 cup granulated sugar, 1 tablespoon flour and the cinnamon in a medium bowl. Halve, pit and thinly slice the plums, then cut each slice in half crosswise.
Cut out small rounds as close together as possible from 1 piece of dough using a 2 1/2-inch round cookie cutter. Gather the scraps and refrigerate until firm; reroll once to cut out more rounds. Meanwhile, arrange 9 rounds on one of the prepared baking sheets and brush the edges with some of the remaining beaten egg. Toss the plums in the cinnamon sugar mixture to coat, then arrange about 4 plum pieces in the center of each dough round on the baking sheet, leaving space around the edges. Top each with a second round of dough and press the edges to seal. (If the dough is too stiff to work with, warm it briefly between your hands to make it more pliable.) Crimp the edges with a fork and cut 4 small steam vents into the top of each cookie. Transfer to the freezer and repeat with the remaining dough and plums.
Lightly brush the cookies with the remaining beaten egg and sprinkle with turbinado sugar. Bake, switching the pans halfway through, until the cookies are golden brown, 22 to 25 minutes. Let cool completely on the pans.