Heat oven to 400ºF. Place a half-sheet pan in the freezer to chill.
Gently unfold the thawed puff pastry sheet and crimp the seams together with your fingertips. Dust each side of the pastry with 1/2 teaspoon of the sugar. Using a rolling pin, roll the pastry in each direction to smooth out the seams. With a sharp pizza cutter, cut out two 6-inch circles of pastry. Place the pastry circles on the chilled half-sheet pan and chill in the refrigerator for 2 to 3 minutes.
Combine the water and lemon juice in a small mixing bowl. Using a vegetable peeler, cut wafer-thin apple slices. Put the apple slices in the lemon juice-spiked water, making sure to coat each slice thoroughly.
Flip the pastry circles over on the half-sheet pan and dock them with a fork to provide an outlet for steam. Put a piece of parchment paper on the pan underneath the pastry circles. Sprinkle the pastry with the remaining teaspoon of sugar and arrange the apple slices on top in an overlapping spiral.
Bake in the middle of the oven until golden brown and crisp, 15 to 20 minutes. Poke the crust; if it feels soft, it needs more time in the oven.
Place the apricot jam in a small bowl and microwave for 30 seconds. Dab (don't brush) the jam on the tops of the tarts. Let the tarts cool for at least 4 hours, then store in a zip-top plastic bag. Serve at room temperature or heat in a 350ºF oven for 10 to 15 minutes and top with ice cream.