Carefully remove rind from orange and from lemon in a continuous strip using a vegetable peeler, making sure to avoid the white pithy part of the rind. Stud orange rind piece with 3 cloves. Stud lemon rind piece with 3 cloves.
Combine Triple Sec, brandy, and cinnamon sticks in a medium saucepan over medium-low heat. Bring to a simmer. Ignite Triple Sec mixture with a long match. Carefully hold orange and lemon rinds with tongs over flames, turning frequently. Let flames die down; add orange and lemon rinds to Triple Sec mixture.
Pour coffee into one side of saucepan. Stir in juices. Discard rinds. Serve in demitasse cups.