Heat a large sauté pan over medium heat, add 2 tablespoons of butter and allow to melt. Add 2 tablespoons of the dark brown sugar and stir until the sugar has melted. Add the apples and stir to combine. Cook until the apples become soft and any liquid they release evaporates, about 10 minutes. Remove from heat and let cool.
In a small bowl, stir together the remaining brown sugar, cinnamon, nutmeg and salt. Melt the remaining butter in another bowl.
Brush the surface of the dough with half of the butter then spread the cinnamon sugar mixture in an even layer, leaving a 1” line down one of the long sides furthest from you. Add the apples over the top, patting them down to ensure they stick. Drizzle with the remaining butter.
Starting with the end closest to you of the long side of the dough, roll the dough up into a tight coil. Cut the log into 9 even slices, about 1¾” thick. This can be done with a knife or using cooking twine.
Butter a 3½-quart baking dish and arrange the dough coils in the dish. At this point the rolls can be covered and refrigerated overnight.
Cover the rolls with plastic wrap and place in a warm place until the dough has risen again, about 1 hour (it may take longer if the dough was refrigerated overnight).
Preheat the oven to 375ºF (190ºC). Bake until the rolls are golden brown and fluffy, about 30 minutes. Let cool for 5 minutes.