Preheat oven to 350°F and line baking sheets with parchment paper.
Cream the butter and brown sugar until light and fluffy.
Beat in the egg and vanilla.
Add the flour, oats, baking soda and salt and mix well.
Gently stir in the blueberries.
Use a medium-sized cookie scoop to form 12-15 cookies.
Bake for 10-12 minutes, or until golden brown; let cool completely on a wire rack.
Make the frosting: Beat the cream cheese and butter until fluffy. Beat in the powdered sugar and vanilla until smooth.
Spread liberally over the cooled cookies.