Preheat oven to 350 degrees F(180 degrees C). Lightly grease cookie sheets or line with parchment paper.
Melt butter and chocolate in small heavy saucepan over low heat; stir to blend. Remove from heat; cool.
Beat egg, brown sugar, vanilla and baking soda in large bowl until light and fluffy. Blend in chocolate mixture and flour until smooth. Slowly beat in milk to make light, cake-batter-like dough.
Drop dough by teaspoonfuls 2 inches apart onto prepared cookie sheets. Bake 10 to 12 minutes or until firm in center. Cut marshmallows in half.
Immediately place halved marshmallow, cut side down, onto each baked cookie. Return to oven 1 minute or just until marshmallow is warm enough to stick to cookie. Remove to wire racks to cool.