Prepare the Filling: Peel, core, and dice the apples. In a medium saucepan over medium heat, melt butter and add the apples, brown sugar, pumpkin spice, cornstarch, kosher salt, and vanilla paste. Cook until the apples soften slightly, and mixture is syrupy, about 3-4 minutes. Let cool slightly before assembling.
Assemble the Turnovers: Roll out the puff pastry on a lightly floured surface. Spoon about 1/4 cup apple filling onto 6 equal sections, leaving a border around the edges. Brush the edges with milk, top with second puff pastry sheet, stretching as needed to cover the filling. Use your fingers to lightly press the puff pastry around each filling section to seal.
Bake: Use a sharp knife to cut into 6 equal rectangles and transfer the turnovers to a parchment-lined baking sheet. Seal the edges with a fork or your fingers. Score tops with a sharp knife. Bake at 400°F (200°C) for 18-20 minutes, or until golden brown and puffed.
Make the Glaze: In a small bowl, whisk together powdered sugar, vanilla, and milk until smooth consistency. Drizzle the glaze over the warm turnovers.