Trim the stems of the beets to about an inch, scrub the beets, and place them in a pot with water to cover. Bring to a boil, and lower the heat, cover and simmer for 30 to 40 minutes, until tender and easily pierced with a sharp knife. Drain and rinse with cold water until cool enough to handle.
Remove and discard the skins, which should slip off easily, and slice the beets into a bowl.
Toss with the lemon juice, vinegar, sesame seeds, onions or scallions, and sugar. Add salt and pepper to taste. Chill thoroughly for about 30 minutes and serve