In a large mixing bowl, combine the melted butter and sugars. Mix well. Add the egg and egg yolk, vanilla extract and red food coloring and mix until smooth.
In a separate bowl sift together the flour, cocoa powder, baking soda, and salt to combine. Add the dry ingredients to the wet and mix together until combined and no streaks of flour remain. Do not over mix.
Use a 1-ounce cookie scoop (or tablespoon) to scoop cookies out. Roll the balls with your hands so they are smooth and place on the prepared cookie sheet an inch apart.
Bake for 10 minutes or until the cookies have puffed up and crackled but are still a little gooey in the center. Do not over bake!
Transfer the cookies to a cooling rack to cool completely before assembling the cookie sandwiches.
While the cookies cool, make the white chocolate frosting. Chop the white chocolate into small pieces and place in a microwave safe bowl. Microwave for 15 seconds and use a rubber spatula to stir the chocolate. Repeat this in 15 second increments until the chocolate is smooth and completely melted (roughly about 45 seconds).
In the bowl of a stand mixer fit with the paddle attachment, add the softened butter, powdered sugar, milk, vanilla extract, and melted white chocolate. Beat on low speed, gradually increasing to high speed.
Mix for 1 minute or until light and fluffy.
Place the frosting into a piping bag fitted with a medium tip.
Pipe roughly 1 tablespoon of frosting onto the bottom of half of the cookies, then sandwich with remaining cookies. If desired, roll the sides in white chocolate shavings. Enjoy!