For the crust, mix 1 egg and 1/2 cup melted butter. Stir in cake mix until combined. May need to mix with your hands as it will be thick. Press evenly into bottom of pan. Bake at 350° for 15 minutes.
In a large bowl, beat cream cheese until fluffy. Add 2 eggs, one at a time, beating on low until combined. Gradually add powdered sugar 1 cup at a time beating on low until smooth. Slowly add remaining 1/2 cup melted butter, chocolate topping, coffee and vanilla. Beat until combined.
Pour over baked crust. Bake 35 minutes; center will not appear to be set and should be a little soft.
Cool pan on rack.
May sprinkle with powdered sugar. Cut into bars; may serve with ice cream. Store in refrigerator.