Combine first 3 ingredients, 2/3 cup sugar, and 4 Tbsp. cocoa in a large bowl. Stir together milk, vegetable oil, and vanilla; add to dry ingredients, stirring just until blended. Spread batter evenly into a lightly greased 8-inch square pan.
Combine chocolate morsels, remaining 1/3 cup sugar, and remaining 2 Tbsp. cocoa. Sprinkle over batter. Bring coffee to a boil in a saucepan; pour boiling coffee evenly over batter. (Do not stir.)
Bake at 350° for 25 to 30 minutes or until cake springs back when lightly pressed in center. Serve warm with ice cream.