Warm 1 cup of milk to body temperature, then pour it into a mixing bowl. Sprinkle in 2 teaspoons of active dry yeast and 2 tablespoons of brown sugar, stirring gently. Allow this mixture to sit for 5-10 minutes until it becomes frothy.

Once the yeast is ready, add ¼ cup softened butter, 2 eggs, 4 cups of all-purpose flour, and 1 teaspoon of salt into the bowl. Mix with a dough hook or wooden spoon for about 4-5 minutes until the dough becomes smooth and elastic.

Transfer the dough into a lightly greased bowl, covering it with a clean kitchen towel or plastic wrap. Place it in a warm area for about 1 hour, or until it doubles in size.

While the dough rises, prepare the cinnamon sugar filling by mixing together 1 cup of brown sugar and 2 tablespoons of ground cinnamon in a small bowl, and grease a 9x13-inch baking dish with butter.

Once risen, punch the dough down gently and roll it out on a generously floured surface into a 12x18-inch rectangle. Spread ¼ cup of softened butter over the dough, then sprinkle the cinnamon sugar mixture and 2 cups of diced Honeycrisp apples on top. Roll the dough tightly from one long side to the other.

Using a sharp knife or dental floss, cut the rolled dough into 12-15 even slices. Arrange these slices in the prepared baking dish, cover with a towel, and let them rise for another 20-30 minutes.

Preheat your oven to 350°F (175°C). Once the rolls have risen, place them in the oven and bake for 25-30 minutes until golden brown.

While the rolls bake, prepare the icing by melting ¼ cup of butter in a small saucepan until golden brown. Then whisk in 4 ounces of cream cheese, ¼ cup of maple syrup, and 1 teaspoon of vanilla extract until smooth.

Once the Apple Cinnamon Rolls are out of the oven and slightly cooled, pour the icing generously over the warm rolls.