To make the caramel sauce, put the sugar and salt in a 2-3 quart heavy-bottomed saucepan. Pour the water over and stir gently. Turn the heat to medium and cook the sugar until it melts into a syrup.
Once the sugar is completely melted, turn the heat up to medium-high and let it cook undisturbed. The sugar will bubble and foam. Continue cooking and keep your eye on the pan just until it the caramel turns deep golden brown and before it starts to smoke, 3-5 minutes.
Immediately remove the pan from the heat. Slowly pour the cream into the pan. The mixture will bubble up intensely so be careful. When the bubbling subsides, stir in the butter until the sauce is smooth. (The sauce can be made ahead and refrigerated. Heat over low heat until it reaches a pouring consistency).
Heat the oven to 400 (200C) degrees.
Roll the puff pastry on a lightly floured surface to flatten slightly. Cut out 4 or 5-inch diameter circles with a pastry cutter or use the upside-down rim of a small bowl as a guide.
Transfer the circles to an ungreased baking sheet. Slice each apple half crosswise into thin slices, keeping them together as you slice. Unfurl the slices onto each pastry circle, fanning them in a pinwheel pattern to cover the dough. Brush each tartlet with melted butter and sprinkle with 1 teaspoon sugar.
Bake 20-25 minutes, until the apples are tender and the edges of the tarts are puffed and golden.
Reheat the caramel to pouring consistency if has cooled, and spoon 1 or 2 tablespoons over each warm tart. Sprinkle with a little flaked salt.
Tip: Use a wooden spoon or a whisk with a heatproof handle to stir the caramel. A metal spoon will get very hot and a plastic utensil will melt.
Extra caramel sauce will keep in the refrigerator for up to 2 weeks.