Cream the butter and sugar together until light and fluffy. Beat in the egg and vanilla.
Whisk together the flour, cocoa powder, baking soda, salt and cornstarch. Stir into the wet ingredients.
Chill the dough for at least 2 hours. Preheat the oven to 350 F. Roll the dough into 1 inch balls and bake for 9-11 minutes or until set. Do not over bake the cookies. They should like slightly doughy when they are taken out of the oven. Gently press 1/2 of a marshmallow on top of each cookie while they are still warm.
Let cool on the cookie sheet for a few minutes and then remove to a wire rack to cool completely.
To make the fudge frosting, add the melted butter, cocoa powder, corn syrup powdered sugar and milk or cream to a microwave safe bowl. Stir together and microwave for about 20 seconds. Stir again. Spoon over cooled cookies and let cool.