In a bowl, beat 2/3 cup powdered sugar and 1/2 cup butter until light and fluffy.
Scrape the sides of the bowl, making sure to mix everything.
Stir in the flour, cinnamon, and vanilla.
Mix until the mixture is the consistency of a thick paste. It will look like wet sand.
Divide into two parts, and place one part in a separate bowl.
If making chocolate conchas, add cocoa powder to one of the bowls, and mix until fully incorporated.
When the dough is done rising, divide into 12 pieces.
Make sure the dough pieces are the same size, or they will not bake evenly.
Shape the dough into balls.
Place on a greased cookie sheet.
Space the dough pieces out about 3 inches apart.
Gently press them down with the back of your hand.
Make 6 balls of the chocolate topping mixture and 6 balls of the white topping mixture.
Place each ball between wax paper.
Using a rolling pin, roll out the balls enough to cover the dough balls.
Grease the top of the dough with melted butter, oil, or shortening.
Place the rolled out topping on top of the dough balls.
Pat down lightly.
Use a knife to cut the grooves in the topping like a shell or criss cross.
Cover and let rise until doubled. About 45 minutes to an hour.
Preheat the oven to 375 degrees F (190 degrees C).
Bake for 20 minutes.