Preheat oven to 375°F.
Whisk together flour, cocoa powder, baking soda and salt in a medium bowl, set aside.
Combine butter and sugar in a large mixer bowl and beat on medium speed until pale and fluffy, scraping down sides as needed. Add corn syrup, egg and vanilla. beat on medium speed for 2 minutes. Gradually add flour mixture alternately with milk until all is well blended.
Drop dough by tablespoon 2-inches apart on baking sheet sprayed with cooking spray. Bake 6 to 8 minutes until set. Remove from oven and immediately press a marshmallow half on top of each cookie. Return to oven and bake an additional two minutes. Cool completely on a wire rack.
Whisk together powdered sugar and cocoa powder in a large mixer bowl. Add butter, milk and vanilla and beat on medium speed 2 minutes or until smooth. Spread about 1 tablespoon of frosting on top of each cookie.