Fudgy Grasshopper Sandwich Cookies

Name: Fudgy Grasshopper Sandwich Cookies
Ingredients: 3/4 cup unsalted butter (6 ounces), at room temperature; 1 cup plus 1 tablespoon granulated sugar , divided; 1 large egg, at room temperature; 1 1/2 teaspoons vanilla extract; 1 1/2 cups all-purpose flour (about 6 3/8 ounces), plus more for dusting and work surface; 2/3 cup unsweetened cocoa (about 2 1/4 ounces); 1/2 teaspoon kosher salt; 4 1/2 tablespoons whole milk; 3/4 teaspoon peppermint extract (such as McCormick); 1 to 2 drops green gel food coloring; 1 1/2 cups white chocolate chips (about 9 ounces); 1/2 cup dark chocolate chips (about 3 ounces); 2 tablespoons whole milk