In a bowl, combine warmed milk, yeast, 1 tbsp melted butter, egg, salt, flour, and sugar. Stir until it forms a ball. If the dough is overly sticky, add additional flour 1 tbsp at a time.
On a well-flour surface, roll dough with a floured rolling pin into a rectangle approximately 12 inches long and 7 inches wide. Dough will be about ¼ inch thick.
Spread softened butter over the dough, leaving a ½ inch perimeter. Sprinkle with brown sugar and ground cinnamon.
2 tbsp butter, 1/3 cup brown sugar, 2 tsp ground cinnamon
Starting with the long edge of the dough furthest from you, tightly roll the dough towards you until you have a solid "roll".
Cut into 9, 1.5 inch rolls. Pro tip: using a bread knife will create the cleanest cuts.
Fit the air fryer basket with foil folded in a way where the edges are up to create a little "pan". Place cinnamon rolls on foil and let rest for 10 minutes.
Bake at 300 degrees F for 10 minutes, checking at minute 8.
Frosting
Combine butter and cream cheese in a bowl and blend with a hand mixer.
2 tbsp cream cheese, 2 tbsp butter
Slowly incorporate powdered sugar and vanilla extract. If you want a pourable frosting, add milk 1 tsp at a time until it is your desired consistency.
Oven-baking: butter a 9x9 baking dish or a pie plate and add cinnamon rolls. Allow to rest for 20 minutes, then bake at 350 degrees F for 23 minutes or until the tops are lightly golden brown.