Trim a small slice off the top and the bottom of each lemon. Standing a lemon on a flat side, cut off the peel and the pith by running your knife from top to bottom, removing the skin (try not to cut into the fruit, but if you do, lay the peel flat and shave off the juicy fruit). Cut the peel into ¼-inch strips. Repeat with the second lemon.
Place the lemon strips in a small pot and cover with cold water. Bring to a boil, then drain. Repeat the process two more times (this helps removes the bitterness from the peel).
In the same pot (now drained), combine 1 cup of the sugar with the water. Bring to a boil, stirring to dissolve the sugar. Add the lemon strips, and return to a boil. Reduce the heat to a simmer and cook until the peels are soft and slightly translucent, about 22 to 25 minutes. In the meantime, pour the remaining ½ cup of sugar into a shallow bowl.
Drain the lemon peels, give them a quick rinse, then drain again. Transfer the peels to paper towels just for a second (to get rid of excess moisture), then plop them into the bowl with the sugar. Using your fingers, gently toss the peels in the sugar to coat. Arrange the candied peels on a piece of wax or parchment paper, and let sit until dry, about 4 to 5 hours.