Preheat oven to 300ºF. Prepare a baking sheet by lining it with parchment paper or a silicone baking mat.
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.
Scoop out the dough by heaping tablespoons and roll them in the sugar* before placing on the prepared baking sheet (about 2 inches apart). Use the back of a round teaspoon to make a small indentation in the cookie dough.
Bake for 10 minutes.
Meanwhile, make the filling: using an electric mixer or small food processor, beat together the cream cheese, egg yolk, sugar, salt, and vanilla until smooth.
After the first 10 minutes of baking, use the back of a round teaspoon to deepen the indentation in the cookies. Spoon a teaspoon of the filling into each indentation. Bake for an additional 12 minutes, until filling is set.
Let cool completely before serving. Store covered and chilled.