Combine whipping cream and 1/3 cup sugar in a small heavy saucepan; bring to a simmer (do not boil). Remove from heat and gradually stir about 1/4 of hot mixture into yolks; add to remaining hot mixture, stirring constantly. Stir in melted sweet chocolate and Kahlúa.
Pour chocolate mixture into 6 (6-ounce) ungreased ramekins. Place ramekins in a 13" x 9" x 2" pan. Add hot water to pan to depth of 1". Bake, uncovered, at 300° for 30 to 40 minutes or just until set. Remove ramekins from pan and let cool on a wire rack. Cover and chill thoroughly.
Combine marshmallow cream, mini-morsels, and milk, stirring until blended. Spoon a small dollop of marshmallow mixture evenly onto each custard.
Sprinkle brown sugar over marshmallow layer. Place ramekins in a 13" x 9" x 2" pan; fill pan with ice cubes, surrounding ramekins.
Broil 3 1/2" from heat (with electric oven door partially opened) 1 to 2 minutes or until brown sugar melts. Remove ramekins from pan. Chill custards 5 minutes or just until brown sugar forms a crust.
Note: Ice cubes keep the custard cold while allowing the brown sugar to melt under heat.