Divide the pastry in half. Pat each half into roughly a square shape about 1 inch thick, wrap each in plastic, and chill for 20 minutes.
Line a rimmed baking sheet with parchment. On a lightly floured surface, using a floured rolling pin, roll out one square of the pastry into a 9x14-inch rectangle. If the dough is too sticky, dust it too with a little flour. Cut the dough into six rounds, each about 4 inches in diameter. Remove the excess dough from around the rounds and discard or save for another use. Run a metal spatula under each round to separate it from the counter.
In a large bowl, stir the 4 tsp. sugar and the flour, cinnamon, and nutmeg. Add the raspberries and gently toss to coat. Taste and add more sugar if the fruit seems tart.
Put a heaping tablespoon of raspberries (three to six berries, depending on size) in a single layer on one half of each dough round. Press gently to flatten the berries a bit. Dampen the pastry edges with a little water and carefully fold the other side of the dough over the berries to make a half moon. Press the edges of the dough together with your fingers or the tines of a fork. If any small cracks formed in the dough, pinch them together as best you can with damp fingers. Use a spatula to transfer the turnovers to the baking sheet.
Repeat this process with the remaining half of the pastry dough and the rest of the berries. When all the turnovers are assembled, refrigerate for at least 15 min. and up to 4 hours. Meanwhile, position a rack in the middle of the oven and heat the oven to 400°F.
When ready to bake, brush the tops of the turnovers (but not the edges or they will get too brown) with the milk and sprinkle with sugar. Bake until golden brown, 20 to 25 min. Transfer to a rack to cool. Serve warm or at room temperature.