Wash the lemon and the orange. Peel the lemon and cut the peel into strips about an inch long and a half inch wide. Stud the skin of the orange with the cloves, inserting the cloves in a spiral pattern from top to bottom. Then cut the peel from the orange in one continuous spiral with the cloves in the center of the strip.
In a metal bowl set over a small burner, combine the brandy and the Cointreau with the lemon peel, orange, and cinnamon sticks. Bring the liqueurs to a light boil and hold it there for a minute. Carefully touch a flame to the mixture and flame it, stirring it around.
With a long fork, spear the orange and hold it up above the bowl. Pour some of the flaming liquid over the orange and let it flow down the spiral of the skin.
Add the vanilla, sugar and the coffee, and swirl it all around until the flames die out. Pour the cafe brulot into demitasse cups and serve hot. It's okay for pieces of lemon or orange peel to go into the cup.