Preheat the oven to 375 degrees F (190 degrees C). Grease cookie sheets. Prepare the filling first. In a 2 quart saucepan, over medium heat, stir together the milk and coconut. Combine the cornstarch and 1/2 cup of sugar; stir into the saucepan. Bring to a simmer and cook, stirring constantly until thickened. Remove from heat and set aside to cool.
Sift together the flour, cocoa, baking soda, cream of tartar and salt; set aside. In a medium bowl, cream together the butter, 1 cup sugar and vanilla until smooth. Stir in the egg and milk. Gradually mix in the sifted ingredients to form a dough. On a lightly floured surface, roll the dough out to 1/4 inch in thickness. Cut into 2 1/2 inch circles with a cookie cutter. Place half of the circles 1 1/2 inches apart onto the prepared cookie sheets. Spoon about 1 teaspoon of the filling onto the center of each cookie on the pan. Top with another circle and seal the edges by pinching.
Bake for 8 to 10 minutes in the preheated oven. Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack to cool completely.
