Melt butter in a small heavy-duty saucepan over medium. Add apricots, and cook, stirring constantly, until slightly softened, 2 to 3 minutes. Stir in strawberries and granulated sugar; cook, stirring constantly and crushing fruit with spoon, 8 minutes.

Stir together flour and lemon juice in a small bowl until smooth. Stir lemon juice mixture into strawberry mixture; bring to a boil. Boil, stirring often, until thickened, about 2 minutes. Remove from heat, and transfer mixture to a small bowl. Cover and chill about 20 minutes.

Meanwhile, preheat oven to 350°F. Roll 1 piecrust into a 12-inch circle on a lightly floured surface. Cut dough into 12 rectangles, using a 2 1⁄2- x 3-inch cutter. Repeat with remaining 3 piecrusts. Place 24 dough rectangles 2 inches apart on 2 parchment paper-lined baking sheets. Spoon about 1⁄2 tablespoon strawberry-apricot mixture into center of each dough rectangle, leaving a 1⁄2-inch border. Dampen edges of dough rectangles with water; top with remaining dough rectangles, pressing edges with a fork to seal.

Stir together egg and milk in a small bowl, and brush tops of hand pies. Cut 2 small Xs in each for steam to escape. Sprinkle each with about 1⁄4 teaspoon Demerara sugar.

Bake in preheated oven until golden, 30 to 35 minutes. Transfer to a wire rack. Serve immediately, or cool completely.