2 and 1/2 cups all-purpose flour
1 teaspoon salt
1 cup (2 sticks) cold unsalted butter, cut into 1/2-inch pieces
1 large egg, beaten
In a food processor, pulse flour and salt to combine. Add butter and pulse until mixture resembles coarse meal, with a few pea-size pieces of butter remaining. Sprinkle with 3 tablespoons ice water. Pulse until dough is crumbly but holds together when squeezed (add up to 3 tablespoons more ice water to achieve desired texture). Do not overmix. Form dough into a 1-inch-thick disk, wrap tightly in plastic, and refrigerate until firm, at least 1 hour.
Preheat oven to 350 degrees. On a lightly floured work surface, divide dough in half and roll out each half to an 1/8-inch thickness. With a 1 and 1/2-inch cookie cutter, cut dough into 36 rounds. Brush 16 with egg and gently press the top 1 inch of a lollipop stick onto each round (if using).
Place 1/4 teaspoon apple butter or applesauce in center of each round and top with another dough round, pressing edges gently to seal.
Transfer pops to a parchment-lined baking sheet. Brush tops with egg and sprinkle with sugar. Freeze on sheet 15 minutes. Bake until golden brown, 25 minutes, rotating sheet halfway through. Let cool on sheet before serving.