Step 1Sift the flour into a mound on the work surface and make a well in the center. Put the egg yolk, salt and 3 tablespoons of water in the well. Pound the butter with a rolling pin to soften it, then add it to the well and work the ingredients in the well with the fingers of one hand until thoroughly mixed.
Step 2Using a pastry scraper, gradually draw in flour from the sides and continue working with both hands until coarse crumbs form. The crumbs should be soft but not sticky; if the crumbs seem dry, sprinkle them with another tablespoon of water. Press the dough gently together into a ball; it will be uneven and unblended at this point.
Step 3Sprinkle the counter with flour. With the heel of your hand, push the dough away from you, flattening it against the counter. Gather it up, press it into a rough ball and flatten it again. Continue kneading until the dough is pliable as putty and pulls away from the counter in one piece, 1 to 2 minutes. Don't overwork the dough or the crust will be tough. Shape it into a ball, wrap and chill in the refrigerator until firm, 15 to 30 minutes.
Step 1While the dough is chilling, peel the apples and cut them in half from stem to bottom; scoop out the cores with a melon baller.
Step 2Melt the butter over medium heat in a 10- to 11-inch oven-proof, straight-sided skillet (preferably nonstick) or a tarte Tatin mold. Sprinkle the sugar over the butter and cook without stirring until the mixture begins to brown and caramelize. The mixture will look coarse and dry at first, but gradually the sugar will melt and brown. Stir gently to avoid splashing sugar on the sides of the pan where it will scorch, and continue cooking until the caramel is deep golden brown, 6 to 10 minutes total. Remove the pan from the heat and let the caramel cool in the pan for 3 to 5 minutes -- the butter will separate, but this does not matter.
Step 3Arrange the apples in the pan in concentric circles standing up on end -- the caramel will help to anchor them. Pack them as tightly as possible because they will shrink during cooking. Cook the apples over medium heat until the juice starts to run, about 8 minutes, then raise the heat and cook them as fast as possible until the undersides are caramelized to deep golden and most of the juice has evaporated, 15 to 25 minutes. If the caramel begins to bubble over the sides of the pan, reduce the heat just enough to keep it cooking quickly without overflowing.
Step 4With a two-pronged fork, turn the apples over, one by one, so the upper sides are now down in the caramel. Continue cooking until these second sides also are golden and almost all the juice has evaporated, 10 to 20 minutes more. The time will vary with the variety and ripeness of the apples, and can take up to an hour in total. When the apples are done, remove them from the heat and let them cool to tepid while heating the oven to 400 degrees.
Step 5Roll the pastry dough to a round just larger than the skillet holding the apples. Wrap the dough around the rolling pin and transfer it to cover the apples. Tuck the edges down around the apples, working quickly so their warmth does not melt the dough. Poke a hole in the center to allow steam to escape. Bake the tart until the pastry is firm and lightly browned, 20 to 25 minutes. Remove the tart from the oven and let it cool for at least 10 minutes, or until tepid. If using a tarte Tatin mold, the tart can be made up to 12 hours ahead and kept in the mold in the refrigerator; if using a skillet, proceed to Step 6 immediately.
Step 6To finish, warm the tart in the skillet on the stove or in the oven before you turn it out; this softens the caramel and loosens the apples. Select a flat platter with a lip to catch any juices; set the platter on top of the tart pan and flip the tart onto the platter. Be careful, because you can be splashed with hot juice. Cut into wedges to serve.