Preheat the oven to 400 degrees F. Line a baking sheet with parchment.
Melt 2 tablespoons of the butter in a large skillet over medium-high heat until it starts to brown. Add the quince and cook, stirring occasionally, until slightly softened and nicely browned in spots, 3 to 4 minutes. Add the sugar and 1/4 cup water, bring to a simmer, and cook, stirring occasionally, until the liquid is syrupy and the quince is tender, 4 to 5 minutes more. Stir in the salt. Transfer the mixture to a metal bowl and refrigerate to cool, about 10 minutes.
Fill a small bowl with water. Dice the remaining 4 tablespoons cold butter into bits and fold it into the fruit. Dust a cutting board with flour and roll out the puff pastry to a 12-inch square. Cut into 4 6-inch squares.
Brush the edges of one of the pastry squares with water. Spoon 1/2 cup of the filling into the center of the square and spread it out a bit towards 2 opposite corners. Fold the dough diagonally over the filling to make a triangle and seal by pressing the edges with a fork. Transfer to the prepared baking sheet. Repeat with the remaining pastry squares and filling. Arrange the turnovers 2-inches apart. Make 2 or 3 vents in the top of each turnover with a sharp knife, using the tip to open the vents to expose the fruit (this will prevent the vents from closing in the oven). If your kitchen is very hot, pop each turnover in the freezer while you work and cut vents them when they are chilled. You can fill and freeze the turnovers for up to 5 days before baking; add an additional 5 minutes to the baking time.
Bake the turnovers until deep golden brown, 18 to 22 minutes, rotating the pan after 10 minutes. Place the pan on a rack and let the turnovers cool for 5 minutes before serving. The turnovers can also be served at room temperature.
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