Preheat the oven to 400˚. Line a baking sheet with parchment paper.

Working with one sheet of puff pastry at a time, unfold the dough on a work surface. Pinch the seams of the dough together.

Cut out 3-inch rounds (about 8 or 9 per sheet) using a cookie cutter or a glass, then punch out a 1-inch round from the center of each using a small cookie cutter or the large end of a pastry tip.

Place the doughnuts (and doughnut holes, if desired) on the prepared baking sheet and bake until browned and crisp, about 20 minutes. Let cool 5 minutes.

Combine the sugar and pie spice in a shallow dish.

Working with one doughnut at a time, lightly brush the tops and sides with melted butter and then roll in the spiced sugar to fully coat. Return to the baking sheet to cool completely.

Meanwhile, for the caramel sauce: Combine the sugar and water in a medium saucepot. Stir to combine, then turn the heat to medium.

Cook, swirling the pan but not stirring, until the sugar is a rich amber color, 8 to 10 minutes.

When the mixture starts to look more like maple syrup than honey, slowly pour in the heavy cream. Stir until the caramel is smooth and has thickened slightly, 1 to 2 minutes.

Remove from the heat, add the butter and salt and stir until smooth. Let cool at least 5 minutes.

Serve the doughnuts with the warm salted caramel sauce for dipping. Store any extra sauce in the refrigerator for up to 1 week.