Peel, halve, and core the apples, being sure to get the stem and flower ends clean. In a heavy ovenproof 10-inch skillet or a tarte Tatin pan, melt the butter, add the sugar and salt, and cook over medium heat until the sugar begins to melt around the edges. Stir gently with a wooden spoon to help the sugar melt completely and then let the mixture cook over medium-low heat until it reaches a deep mahogany-colored caramel; if using a dark pan, be sure to lift some caramel out with a spoon to check the color. The butter will pool on the top-that's fine. As soon as the caramel is the right color, take the pan off the heat.
Set aside four of the apple halves and arrange the remaining ones in the pan in concentric circles, standing them upright on their narrowest end and packing them in as tightly as possible (they'll shrink as they cook). Try to make the rows look neat.
Return the pan to medium heat, increasing to medium high as the apple juices begin to flow-the liquid should bubble gently. Rotate the pan occasionally in case there are any hot spots. Use your sense of smell: If you start to smell burnt sugar, immediately turn down the heat. When the bottom halves of the apples are caramelized and slightly tender, flip each one over with a fork and continue cooking.
About this time, there should be a lot of juice in the pan and the apples will likely have shrunk and be slumping a bit. Carefully take the pan from the heat and, holding the apples back with a spatula, pour off 1/2 to 1 cup of caramel and juice into a small saucepan (pour off more if necessary; you should have about an inch of liquid left in the skillet). Return the skillet to the heat and continue cooking. Add the reserved apples to the caramel in the smaller pan and cook over medium-high heat until they're caramelized, about 10 minutes, turning them frequently as necessary.
With a rubber spatula or a wooden spoon, push the apples in the skillet so they're tightly packed and upright again; there will be gaps. Holding them in position, transfer an apple half from the small pan and slide it, wider end down, into a gap; repeat with as many apple halves as will fit nicely; reserve the juice in the pan. Continue cooking the apples in the skillet until they're tender and thoroughly caramelized, inserting another apple half if necessary. The total cooking time could take 35 minutes or more.
Take the skillet off the heat and let cool to room temperature. If there's more than about 1/2 inch of liquid remaining in the pan, carefully pour off the excess into the small saucepan.
Heat the oven to 375°F. Roll the chilled pâte brisée dough into a round about 1 inch larger than the pan; the dough should be about 1/8 inch thick. Roll the round of dough onto the rolling pin and transfer it to the skillet, gently draping the dough over the apples. Tuck and fold the edge of the dough under to make a rim. Put the skillet in the middle of the oven. (Put a baking sheet on the rack below to catch any overflowing juices.) Bake until the crust is a rich brown and looks crisp, about 25 minutes.
Take the tarte out of the oven and let it cool for about 15 minutes. If more juice accumulates, carefully pour off most of it into the saucepan and then invert a large plate on top of the pan, flip the pan and plate over in one quick move, and lift off the pan. Simmer the reserved caramel and juices until thick and syrupy and then spoon them over the finished tarte or serve alongside. Serve the tarte warm or at room temperature with crème fraîche or vanilla ice cream. Leftovers are best eaten within a day or so; don't refrigerate.