In a medium saucepan, combine the water, vinegar, sugar (if using), salt, and sliced beet and bring to a simmer. Cover with a lid and simmer until the beet is fork tender, about 20 minutes. Transfer liquid and beets to a large bowl or mason jar, and allow to cool completely.
Once cool, place eggs in bowl/jar. Cover and place in the fridge. Eggs will be ready after 2-3 hours, but for pinker eggs with a more pickled flavor, allow them to sit in juice for 2-3 days. Enjoy!