Thaw puff pastry on a baking sheet lined with parchment paper for 30 minutes.
Meanwhile, peel, core, and thinly slice the apples (about 1/8").
Unfold thawed pastry and lightly roll to smooth.
Score a 1/2" border around the edge with the back of a knife. Prick the center all over with a fork.
Sprinkle almond flour over the pastry, avoiding the borders.
Arrange apples in 4 rows (or 3 if the apples are larger), using similar-sized slices, fanned slightly and stacked for height. Use one of the smaller slices behind the first row to hold it up if needed.
Mix sugar and cinnamon, then sprinkle over the apples. Dot with small pieces of butter.
Bake at 400°F for 40 minutes, until the edges are golden and the apples are tender.
Cool slightly. Warm apricot preserves with 1 tbsp water in the microwave for 20 seconds, then brush over the apples and edges of the pastry.
Serve slightly warm or at room temperature.