Prep. Preheat the oven to 425°F. Lightly butter the muffin cups, including the sides. Cut 12 rounds from the pastry with the cutter, lay them on a plate and chill. Toss the apples in the lemon juice.
Soften the apples. Melt the butter in the lidded saucepan over medium heat. Add the apples, cover and cook for 5 minutes, until just tender. Remove from the heat and set aside while you make the caramel.
Make the caramel. In the second saucepan, stir the sugar, water and lemon juice over medium heat until the sugar dissolves. Once it starts bubbling, stop stirring and swirl the pan occasionally until the caramel turns deep amber. Remove from the heat and stir in the butter and salt. Working quickly, spoon a little caramel into each muffin cup.
Fill. Divide the apples among the cups, packing them in snugly. For a prettier finish, use thin slices and fan them in an overlapping circle.
Top. Lay a pastry round over each cup and tuck the edges down around the apples. Prick each top once or twice with a fork.
Bake. Set the muffin pan on the baking sheet to catch any drips. Bake for 18 to 22 minutes, until the pastry is deep golden and puffed.
Flip. Let rest for 5 minutes. Lay the baking sheet (wiped clean) over the pan and, wearing oven mitts, flip it in one quick motion. Lift the pan away and press any stray apples back into place.
