Instructions

Adjust oven rack to middle position and heat oven to 400 degrees. Line rimmed baking sheet with parchment paper. Melt 2 tablespoons butter in 12-inch skillet over medium-high heat. Add sage and continue to cook, swirling skillet occasionally, until butter is dark golden brown and has nutty aroma, about 2 minutes. Transfer browned butter mixture to large bowl; set aside.
Melt remaining 2 tablespoons butter in now-empty skillet over medium heat. Add squash, apple, shallot, sugar, salt, and pepper; cover; and cook, stirring occasionally, until squash and apple are softened and lightly browned, 13 to 16 minutes. Off heat, stir in vinegar and, using potato masher, mash half of mixture until mostly smooth. Add to bowl with browned butter mixture and stir to combine. Let cool completely, about 30 minutes, then stir in ¾ cup Gruyère. (Filling can be refrigerated for up to 24 hours.)
Dust counter lightly with flour. Working with 1 piece puff pastry at a time, unfold pastry and roll into 10-inch square. Cut into four 5-inch squares. Space pastry squares evenly on prepared sheet. Divide squash mixture evenly among pastry squares, mounding about 3 tablespoons in center of each square, then brush edges of pastry with egg wash (reserve remaining egg wash). Fold pastry over filling to form triangle. Using fork, crimp edges of pastry to seal. Cut two 1-inch slits on top of each turnover (do not cut through filling). (Unbaked turnovers can be frozen in airtight container for up to 1 month.)
Brush tops of turnovers with remaining egg wash and sprinkle with remaining ½ cup Gruyère. Bake until well browned, 22 to 25 minutes, rotating sheet halfway through baking. Let turnovers cool on wire rack for 15 minutes. Serve warm or at room temperature.