Root Beer Float Sandwich Cookies

Name: Root Beer Float Sandwich Cookies
Ingredients: 226 grams (2 sticks) unsalted butter, at room temperature; 400 grams (2 cups) light brown sugar; 2 large eggs, at room temperature; 2 1/2 teaspoons Root Beer Concentrate (NOT extract!); 2 teaspoon vanilla extract; 450 grams (3 1/2 cups) all-purpose flour; 1 teaspoon baking soda; 1 teaspoon kosher salt; 1/4 cup buttermilk (or make your own - tip below); 56 grams (4 Tablespoons) unsalted butter, at room temperature; 1 Tablespoon shortening; 250 grams (2 cups) powdered sugar; 1/2 teaspoon kosher salt; 1/2 teaspoon Root Beer Concentrate; 1 teaspoon Vanilla extract; 1 to 2 Tablespoons milk