In a large bowl, cream the butter and sugar together until light and fluffy (2-3minutes).
Add eggs and mix well. Add the root beer extract and continue to mix.
In a separate bowl, mix the flour, baking soda and salt.
Add the flour mixture to the butter/sugar mixture and mix until well combined. Add the milk, one tablespoon at a time until the mixture is not crumbly. The dough should just be a little bit sticky. Cover the cookie dough and chill for at least one hour.
Preheat the oven to 350°. Use a cookie scoop to scoop the dough into balls and place them on a prepared cookie sheet. Just barely press down the tops of the cookie dough balls.
Bake the cookies for 8 minutes. Let them cool on the cookie sheet for a few minutes before cooling them on a wire rack. Once the cookies are completely cooled, frost with Root Beer Buttercream frosting.
Root Beer Buttercream frosting
Use an electric mixer to beat the butter for 2-3 minutes, until fluffy.
Add powdered sugar and mix until smooth.
Add the root beer extract and mix well.
Add milk to bring frosting to desired consistency.