Cut the peels of the lemons and oranges in 1/4 inch thickness. Make sure you remove most of the pits from them.
In a medium sized saucepan, place the lemon and orange peels and pour in enough water to cover them. Bring to a boil over high heat. Reduce heat and simmer for 20 minutes, drain and set aside.
In a heavy-bottomed saucepan, add 2 cups sugar and 1 cup water. Bring to a boil and cook until the mixture reaches the thread stage. To check when has it reached the thread stage, drop a small amount in cold water - if it forms a soft thread; it is done.
If you have a candy thermometer, bring it to 108 degrees C. Add in the boiled peels, reduce heat and simmer for about 5 minutes. Drain the excess water with a sieve.
Roll the peel pieces, a few at a time, in the 1/3 cup of sugar that was set aside for decoration. Let the pieces dry on a wire rack. They are ready for consumption as soon as they are of room temperature. Store in an airtight container for later use.