In a large stockpot, combine 10 cups water and 1 cup sugar. Using a vegetable peeler, remove the zest from 1/2 lemon, and add to the pot. Halve the lemons, and juice 1 1/2 of them directly into the pot; set the other half aside.
Peel, halve, and core the quince, and add to the pot. Bring to a boil. Cover, reduce to a simmer, and cook until tender, but not completely cooked, 8 to 10 minutes. Drain, and set aside.
Preheat oven to 375 degrees. In a 10-inch tarte Tatin pan or cast-iron skillet, combine remaining 3/4 cup sugar and the salt. Place over medium heat, and cook until it begins to thicken and turn a golden brown. Remove from heat, and stir in butter.
Arrange the quince in the pan, cut sides up, in a slightly overlapping circular pattern. (Remember that since the tarte Tatin is inverted after it is cooked, the fruit on the bottom will be visible when served.) Place a few quince halves in the center, cut side up. Sprinkle with the juice of the remaining lemon half.
On a lightly floured work surface, roll dough out into a 12-inch round, about 1/8 inch thick. Place dough over quince, tucking the edges of the dough around the fruit. Bake until juices are bubbling and crust is golden brown, about 45 minutes. Remove from oven, and let stand for 10 minutes. Using a paring knife, loosen the pastry from the edges of the pan. Place a serving platter over the pan, and carefully invert. Serve warm.