How to Make It
In a 9-inch cast-iron skillet, melt the butter and brown sugar over moderate heat, stirring to combine. Turn off the heat.
Fit the apple quarters on their sides snugly in the skillet. Cook over moderately high heat until the juices are bubbling, about 5 minutes. Turn the apples with a spoon and cook for 5 minutes longer. Turn them rounded side up, reduce the heat to moderate and cook for 5 minutes, then turn them rounded side down and cook for 4 to 5 minutes longer. The apples should be soft when pressed gently but not falling apart. Remove the skillet from the heat.
Process the almonds and sugar in a food processor until finely ground. Add the flour and salt and pulse to combine. Add the butter and pulse until the mixture resembles coarse meal. Add the egg and pulse briefly to incorporate. Add the water 1 teaspoon at a time and pulse until the dough can be gathered into a ball. Transfer it to a lightly floured sheet of wax paper and, with floured fingers, pat the dough into an even 9 1/2-inch round. Slide the paper with the crust onto a cookie sheet and refrigerate until firm or for up to 4 hours.
Preheat the oven to 400°. Let the dough stand at room temperature for about 10 minutes. Peel off the wax paper and cover the apples with the dough round. Bake for about 25 minutes, or until the pastry is golden. Let sit for 10 minutes, then invert the tart onto a large flat platter and serve it warm.