Toss the apples, raspberries, granulated sugar and lemon juice in a large bowl. Melt the butter in a medium skillet over medium heat. Add the fruit mixture and cook, stirring occasionally, until the apples are soft and the raspberries break down, 8 to 10 minutes. Sprinkle in the flour, cinnamon and nutmeg and stir until the juices begin to thicken, about 1 minute; set aside to cool completely.
Preheat the oven to 375 degrees F and line 2 baking sheets with parchment paper. Unroll the pie dough on a lightly floured surface. Cut out 2 1/2-inch rounds with a cookie cutter or glass (you should get about 48 rounds). Arrange half of the rounds on the prepared baking sheets; put a rounded teaspoonful of filling in the center of each. Brush the edges of the dough with water, then top with the remaining dough rounds, crimping the edges together with a fork to seal.
Brush the pies with the beaten egg and sprinkle with turbinado sugar. Cut a small X in the center of each to allow steam to escape. Transfer to the oven and bake until golden, 22 to 24 minutes. Let cool slightly on the baking sheets, then insert a lollipop stick into each. Let cool completely. Store in an airtight container up to 3 days.
Courtesy of Food Network Magazine, Photograph by Charles Masters