Scoop 1 tbsp sized balls of Nutella onto a plate lined with parchment paper. Put in the freezer and let chill / harden while you make the dough.
Preheat the oven to 400 degrees. Cream together butter, brown sugar, and white sugar for 2 minutes. Add eggs, vanilla, and red food coloring and mix again.
Add flour, cocoa powder, cornstarch, baking soda, and salt. Mix until combined. The dough should be soft, but not too sticky.
Take the Nutella out of the freezer. Using a large cookie scoop, fill the scoop most of the way, then press a large hole in the center with your fingers. Fill the hole with a chilled chunk of Nutella, then close the dough around the Nutella.
Put up to six dough balls on a cookie sheet at bake at 400 degrees for 9-11 minutes.
While the cookies are cooling, make the frosting. Combine the cream cheese and butter in a bowl and mix for 2-3 minutes. Add vanilla and powdered sugar and mix for an additional 2-3 minutes. Add frosting to a piping bag.
Once cookies are completely cooled, pipe on frosting and enjoy!!