Step 1: Prepare the Dough

1. In a small bowl, mix warm milk, yeast, and 1 tablespoon of sugar. Let sit for 5-10 minutes until foamy.
2. In a large mixing bowl, whisk melted butter, egg, egg yolk, and remaining sugar. Add the yeast mixture and stir to combine.
3. Gradually mix in flour and salt, kneading until a soft, slightly sticky dough forms.
4. Transfer to a lightly greased bowl, cover with a damp towel, and let rise for 1-1.5 hours, or until doubled in size.

Step 2: Make the Apple Pie Filling

1. In a saucepan, melt butter over medium heat. Add apples, brown sugar, granulated sugar, cinnamon, nutmeg, and vanilla.
2. Stir and cook for 5-7 minutes, until apples are soft.
3. Mix cornstarch with 1 tablespoon of water, add to the apples, and stir until thickened. Remove from heat and cool.

Step 3: Assemble the Cinnamon Rolls

1. Once the dough has doubled, roll it out into a 12×18 inch rectangle on a lightly floured surface.
2. Spread the cooled apple pie filling evenly over the dough.
3. Roll the dough tightly from the long edge to form a log. Slice into 12 even rolls using a sharp knife or dental floss.
4. Arrange the rolls in a buttered 9×13-inch baking dish, cover, and let them rise for 30 minutes.

Step 4: Bake the Rolls

1. Preheat your oven to 350°F (175°C).
2. Bake for 25-30 minutes, or until golden brown on top.

Step 5: Make the Brown Butter Maple Icing

1. Melt butter in a saucepan over medium heat until it turns golden brown and smells nutty. Remove from heat and cool slightly.
2. In a mixing bowl, beat cream cheese, maple syrup, and vanilla extract until smooth.
3. Add powdered sugar and browned butter, whisking until creamy.

Step 6: Serve and Enjoy!

1. Drizzle warm cinnamon rolls with brown butter maple icing.
2. Serve immediately and enjoy the gooey, sweet, cinnamon-apple goodness!