Beat white sugar and 1 cup butter together with an electric mixer in a large bowl until creamy. Add eggs 1 at a time, beating well after each addition. Beat sour cream and 2 teaspoons vanilla extract into butter mixture.
Stir flour, baking powder, and baking soda together in a bowl. Add flour mixture to sour cream mixture and stir well. Cover bowl with plastic wrap and refrigerate for at least 2 hours or overnight.
Preheat oven to 350 degrees F (175 degrees C). Lightly grease 2 baking sheets.
Roll cookie dough out on a floured surface to 1/4-inch thick. Cut into 3-inch round cookies and place onto prepared baking sheets.
Bake in the preheated oven until cookies spring back when lightly touched in the center, 10 to 12 minutes; transfer to wire racks to cool completely.
Beat 1/3 cup butter, confectioners' sugar, milk, 1 1/2 teaspoons vanilla extract, and salt together in a bowl until smooth. Spread frosting over cooled cookies.