Preheat oven to 350 degrees and line a baking sheet with parchment paper.
In a small bowl, mix the egg replacer with water- follow directions on the package. Set aside.
In a medium bowl, combine flour, cocoa, baking soda, and salt. Whisk and set aside.
Using a hand or stand mixer, combine the butter and sugar until creamed. Add egg replacer. Mix together. Stir in vanilla. Pour in the flour mixture a little at a time until completely incorporated.
Wrap the dough in plastic wrap and let chill for 1-2 hours.
Roll the dough into 1 1/2 inch balls. Pour 1/3 cup sugar into a small bowl and roll the dough around until coated.
Bake for 12 minutes. Let cool for a few minutes and then transfer to a cooling rack.