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Preheat the oven to 400°. Line a baking sheet with parchment paper. On a lightly floured work surface, roll out the puff pastry to a 12-by-10-inch rectangle, about 1/8 inch thick. Transfer the pastry to the baking sheet and brush with 1 tablespoon of the melted butter.
Arrange the apple slices on the pastry in 3 neat lengthwise rows, overlapping them slightly and leaving a 1/2 -inch border all around. Brush the apples with the remaining 1 tablespoon of melted butter and sprinkle with 1 tablespoon of the sugar. Bake the tart on the bottom rack of the oven for about 30 minutes, or until the pastry is cooked through and nicely browned. Remove from the oven and preheat the broiler.
Sprinkle the apples with the remaining 1 tablespoon of sugar and broil for about 30 seconds, or until the apples start to brown on top; rotate the pan as necessary. Cut the apple tart into 6 large squares and serve hot.