In a large bowl, beat butter on high until fluffy.
With the mixer on medium, add in the sugar and salt. Beat on high until light and fluffy (about 2 minutes).
Add in the egg and vanilla and mix until completely combined.
Slowly add in the flour until combined. Scraping the edges of the bowl as needed.
Cover the bowl with plastic wrap and chill for a minimum of 30 minutes.
While the dough is chilling, make room in the freezer to freeze the "cinnamon roll" later on.
Once dough is easy to handle, flour a clean surface and roll dough into a 15x10 inch rectangle.
Brush the rectangle dough with the lightly beaten egg.
In a separate small bowl, combine brown sugar and cinnamon.
Sprinkle the brown sugar mixture all over the cookie dough rectangle.
Roll up the rectangle from the long side.
Place the roll on a cookie sheet and place in the freezer for 30 minutes or until firm.
Preheat oven to 375 F degrees.
Remove cookie roll from freezer and slice into 1/4" slices.
Place cookies on a parchment lined or nonstick cookie sheet.
Bake for 8-10 minutes or until edges are golden in color.
Remove from baking sheets and transfer to wire rack. Let the cookies cool completely before icing.
Using a hand mixer and medium bowl, mix together cream cheese and butter until smooth.
Beat in the powdered sugar until combined.
Add the milk until the desired consistency is reached.
Ice the cookies.