Preheat the oven to 400 degrees. Line 2 sheet pans with parchment paper.
Cut each sheet of puff pastry into 4 squares. Divide the pastry between the prepared sheet pans and refrigerate while you prepare the apples.
Peel the apples and cut them in half through the stem. Remove the stems and cores with a sharp knife and a melon baller. Slice the apples crosswise in 1/4-inch-thick slices. Place overlapping slices of apples diagonally across the pastry and place one slice of apple on each side of the arranged slices. Sprinkle the apples evenly with the sugar and dot them with the butter.
Bake for 40 minutes, until the pastry is browned and the edges of the apples start to brown. Don't worry! The apple juices will burn in the pan but the tarts will be fine! When the tarts are done, heat the apricot jelly together with the Calvados until bubbly and brush the apples and pastry completely with the jelly mixture. Loosen the tarts with a metal spatula so they don't stick to the paper. Allow to cool and serve warm or at room temperature.
2012, Ina Garten, All Rights Reserved
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