Preheat the oven to 400°F. Lightly grease the inside of a 12-cup muffin tray set on a rimmed baking sheet.
Place the puff pastry on a lightly floured surface and use a 3-inch round cookie cutter to cut out 12 circles. Place the circles on a tray and prick pastry all over with a fork. Chill.
Pour 1/2 cup water into a medium saucepan over medium heat. Add the sugar and honey and increase the heat to high. Swirling the pan occasionally, cook until the sugar is dark amber, 10 to 12 minutes. Take the pan off the heat and whisk in the butter one piece at a time until completely emulsified. Divide the sauce among the muffin cups. Divide the apple slices among the muffin cups, breaking them in half when necessary to pack all the slices in.
Cap each muffin cup with a circle of puff pastry dough. Bake for 20 minutes, then reduce the temperature to 350°F and continue baking until the pastry is golden-brown and the apples are tender when pricked with a fork, another 20 to 25 minutes.
Transfer the muffin tray to a wire rack to partially cool for 20 to 30 minutes. Use a spoon to help scoop out the tarts, invert them (so that the apple side is up) onto a platter. Or, you can let tarts cool and reheat in a 350°F oven for 5 to 10 minutes before inverting onto serving dish.