To prepare the dressing, combine the first 8 ingredients, stirring with a whisk. Set aside.
To prepare the salad, arrange the romaine in the bottom of a 2-quart glass bowl or trifle dish; top with iceberg. Layer radicchio, endive, cucumber, celery, carrot, bell pepper, tomato, red onion, and green onions evenly over lettuces. Drizzle with dressing. Cover and chill 4 hours or overnight.
Arrange beets over green onions; top with eggs. Garnish with parsley, if desired.