On a cutting board, roll the pie crust into a 10-inch circle. Use a pizza cutter or sharp knife to cut the pie crust into roughly 2-inch square pieces. (Some pieces will not be perfect squares.) Place the pieces of crust on a parchment-lined baking sheet and place them in the freezer while you make the filling.
Preheat the oven to 400°F.
In a large bowl, combine the apples, brown sugar, lemon juice, cinnamon, and salt.
Place the butter in a 10-inch cast-iron or ovenproof skillet over medium heat. Cook for 3 to 4 minutes until melted. Add the apple mixture and cook for 6 to 8 minutes, stirring occasionally, until the apples start to soften and their juices are bubbling.
Meanwhile, in a medium bowl, whisk to combine the apple juice (or water) and cornstarch. Add this mixture to the skillet and immediately mix well to combine. Cook for 1 more minute, just until the liquid around the apples starts to thicken. Remove from the heat.
Use the back of a spoon to spread the apple mixture into an even layer in the skillet. Place the pie dough pieces randomly over the top, shingling them slightly while leaving a little bit of space between some pieces where steam can release. Brush the crust all over with heavy cream or milk and sprinkle with coarse sugar.
Bake for 15 minutes. Remove from the oven and press down on the crust with the back of a spoon in 6 to 7 places all over the top so that the pan juices flow over the crust in certain spots. You do not want to submerge the whole crust! Just about half of it.
Return to the oven for about 15 more minutes until the crust is golden brown. Let cool for 15 minutes before serving.
Notes
Best sugars for sprinkling: Turbinado or sparkling sugar are great coarse sugars for a little crunch. You could also use cinnamon sugar or a different spiced sugar for additional flavor.
If you don’t have an ovenproof skillet, you can transfer the apple mixture to a 2½ to 3-quart casserole dish before topping it with the crust and baking.
If you don’t have heavy cream or milk, you can also brush the crust with an egg wash (which is just 1 egg whisked with 1 tablespoon of water).
If you don't have cornstarch, you can sprinkle 2 tablespoons of all-purpose flour over the cooked apples, then stir in the apple juice or water. Cook until the mixture simmers and thickens.