Cream butter and sugar together until light and fluffy, about 2 minutes. Scrape down bown and beat in egg, egg yolk and vanilla until completely smooth. Add flour and baking powder and mix until just combined. Cover dough and chill for at least 1 hour.
Preheat oven to 350. Grease or line 2 baking sheets with parchment paper, set aside. Take dough out of refrigerator and using a cookie scoop, scoop out 2 tbs of dough, roll into a ball and flatten with your hands or place on cookie sheet and flatten with the bottom of a flat glass. Bake cookies for 9-13 minutes or until the cookie edges are set. Bottoms should be just barely browned. Allow to cool on baking sheet for 5-7 minutes then transfer to a wire rack to finish cooling.
While cookies are cooling, prepare frosting. Cream butter, sugar and salt together until light and fluffy, about 2 minutes. Add in vanilla and 1/2 tsp of cream/milk until smooth. Beat for an additional 2-3 minutes and add more liquid until desired consistency is reached.