Preheat oven to 300ºF. Prepare a baking sheet by lining it with parchment paper.
Beat together the butter, brown sugar, egg yolk, and vanilla until light and fluffy. Mix in the flour, food coloring, and cocoa powder until soft dough forms.
Pinch about a tablespoon of the dough at a time roll into a ball then roll them in the white sugar before placing on the prepared baking sheet (about 2 inches apart). Use the back of a small spoon to make a small indentation in the cookie dough.
Bake for 10 minutes. Meanwhile, make the filling: using an electric mixer or beat together the cream cheese, egg yolk, sugar, salt, and vanilla until smooth.
After the first 10 minutes of baking, remove from the oven. Then use the back of a round teaspoon to deepen the indentation in the cookies. Spoon a teaspoonish of the filling into each indentation.
Bake for an additional 12 minutes, until filling is set. Let the cookies cool completely before serving. Store in a air tight container and chilled.