Begin by making the cinnamon rolls. In a large bowl, combine warm milk and yeast, and let it sit for about 5 minutes until the yeast blooms.
Add maple syrup, melted butter, egg, and vanilla to the milk mixture, and whisk until well combined.
Using a stand mixer with a dough hook attachment, mix flour and salt into the wet ingredients until a smooth dough forms. Knead for 2-3 minutes until elastic.
Transfer the dough to an oiled bowl, cover with plastic wrap and a towel, and let it rise in a warm place for 1-1.5 hours.
While the dough rises, prepare the apple filling by combining brown sugar and cinnamon in a small bowl.
Peel and dice the apples, then sauté them with 1 tablespoon of butter and nutmeg until soft.
For the brown butter cream cheese frosting, brown 3 tablespoons of butter in a pot, let it cool, then mix it with cream cheese, vanilla, cinnamon, and powdered sugar until smooth.
Roll out the risen dough into a rectangle, spread softened butter over it, sprinkle with the cinnamon-sugar mixture, and top with sautéed apples.
Roll the dough into a log and slice into 1-inch rolls.
Place the rolls in a buttered or parchment-lined baking dish, cover, and let rise for 30 minutes.
Preheat the oven to 350°F. Bake the rolls for 18-22 minutes until set.
Let the rolls cool for 10 minutes before frosting with the brown butter cream cheese mixture.