In a small saucepan, heat the milk and apple cider to 110 degrees over medium low heat, then pour them into the bowl of a stand mixer with the hook attachment.
Add 2 tablespoons of the granulated sugar and the yeast package. Let the mixture sit until it starts to foam, about 5 minutes.
Add the remaining sugar, softened butter, and egg and mix at low speed for about 1 minute.
Add the flour and salt to the mixture and mix until the dough comes together for about 2-3 minutes.
Continue to mix/knead for another 5 minutes. The dough should be soft and a bit tacky but not stick to the sides of the bowl. If the dough is too sticky, add one teaspoon of flour.
Remove the dough from the bowl and shape it into a ball. Place in a large bowl that has been lightly oiled (no more than 1 tablespoon of vegetable oil in the bow). Cover with plastic wrap or a kitchen towel and let rise at room temperature until doubled in size, which could take 1 ½ to 2 hours, depending on the weather.
Grease the bottom of a large cast iron pan or 9x13 baking dish.
Punch down the dough and turn it out onto a lightly floured surface. Using a rolling pin, roll the dough out until it is about ½” thick and about a 12” x 18” rectangle.
Mix the brown sugar, cinnamon, and chopped apple in a small bowl. Set aside.
Spread the softened butter over the dough, then top it with the apple sugar mixture, leaving a small amount of space along the long edge of the dough to start the roll.
Roll up the dough tightly and then cut into 12 pieces.
Arrange the rolls in the greased cast iron skillet or baking dish, loosely top with plastic, and let rise for about 30 minutes.
Preheat the oven to 375 degrees while the dough is rising.
Bake for 28-30 minutes until they are lightly browned on top.
While baking, mix the icing by placing the butter and cream cheese in a bowl and mix with an electric mixer until creamy, about 2 minutes. Add in the milk, vanilla. Then, slowly add the powdered sugar until smooth and fluffy.
Let the cinnamon rolls cool for 10 minutes, then spread icing over the rolls and serve warm.