In a stand mixer bowl, combine 1.5 cups flour, sugar, instant yeast, and salt.
Add the warmed milk, 2 tablespoons softened butter, and egg; mix on low speed using the dough hook attachment until combined.
Gradually add remaining flour while kneading for 5 to 7 minutes until dough is smooth and clears the sides of the bowl.
Cover dough and allow to proof in a warm environment (approx. 80°F) for 30 minutes.
Roll dough on a floured surface into a 12x8 inch rectangle approximately 0.25 inches thick.
Spread 0.25 cup softened butter over the dough, leaving a 0.5-inch border, and sprinkle with combined brown sugar and cinnamon.
Roll the dough tightly starting from the long edge and slice into 8 equal rounds using unflavored dental floss or a serrated knife.
Line the air fryer basket with perforated parchment paper and arrange rolls with 1 inch of clearance between them.
Set the air fryer to 320°F (160°C) and cook for 8 to 10 minutes until the internal temperature reaches 190°F.
Whisk cream cheese, powdered sugar, 1 tablespoon milk, and vanilla until homogenous; apply to rolls while internal temperature is above 140°F.