Make the cinnamon rolls. Add the warm milk and yeast to a large bowl. Set aside until the yeast has bloomed-- about 5 minutes.
Next add the maple syrup, melted butter, egg, and vanilla. Whisk to combine.
Using a stand mixer on low-medium speed with the dough hook attachment, slowly stir the flour and salt into the wet ingredients. Once all of the flour has been added and a shaggy dough has formed, knead for 2-3 minutes or until the dough is smooth and elastic (it should bounce back when you poke it).
Transfer the dough to a lightly oiled bowl and cover it with plastic wrap and then a tea towel. Place it in a warm place to rise until doubled in size -- about 1-1.5 hours. While the dough rises you can make the filling and cook your apples (below).
Make the apples. In a small bowl, combine the brown sugar and cinnamon. Set aside until ready to use.
Next, peel your apples and dice them into small pieces.
Then, heat a sauté pan over medium heat and begin to melt 1 tablespoon of butter. Once the butter is melted, add your diced apples and nutmeg, and cook for 5-7 minutes or until the apples are soft. Set aside.
Make the brown butter cream cheese frosting. In a small pot or skillet, brown 3 tablespoons of butter over medium heat, stirring continuously. Let cool.
Combine the cream cheese, vanilla extract, cinnamon, and browned butter in the bowl of a stand mixer fitted with the paddle attachment.
When the cream cheese and butter are smooth, turn the mixer to low speed and slowly add the powdered sugar.
Scrape down the sides of the bowl and turn the mixer to medium speed. Beat for 30-45 seconds or until the frosting is light and fluffy. Set aside.
Assemble the cinnamon rolls. Once the dough has risen for 1 to 1.5 hours, turn it out onto a lightly floured surface.
Roll it into a rectangle that is approximately 18” long, 12” wide, and 1/4” thick.
Spread the butter over the dough and then sprinkle the filling over the butter, and then sprinkle the diced apples over the filling. Then roll into a tight log, starting at one of the long sides. For larger rolls, roll the tight log starting at one of the short sides.
Slice the log into 1” rolls. This will yield about 12 rolls, depending on how tightly you roll your log and whether you start rolling at a long or short side of the dough.
Butter a 9x13-inch baking dish or line it with parchment paper. Add the rolls to the dish, making sure they just barely touch. Depending on how tightly you rolled dough into the log, you can either place rolls 3x4 OR 2x4 with extra rolls in another baking dish.
If you’re making the rolls in advance, wrap them in plastic and place them in the fridge for up to 12 hours. Bring the rolls to room temperature (about 30 minutes) when you’re ready to bake and then skip to step 13.
If you aren’t making the rolls in advance, cover with plastic wrap and then a tea towel, and place them in a warm spot to rise until the rolls begin to touch -- about 30 minutes. While the rolls rise, preheat your oven to 350°F.
Bake the apple cinnamon rolls for 18-22 minutes (depending on your oven) or until the centers are set, be sure not to overbake.
Allow the cinnamon rolls to cool for 10 minutes before frosting.