Place potatoes, beets, and carrots into a large saucepan and cover with water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes. Drain and let cool, at least 15 minutes.

Heat oil in a skillet over medium heat. Add onion; cook and stir until softened, about 5 minutes.

Peel potatoes, beets, and carrots. Coarsely grate vegetables and eggs into separate, individual bowls. Season with salt and pepper.

Place a small glass in the middle of a serving plate. Layer salad around the glass into a bracelet-like shape in the following order: potatoes, 1/2 the beets, carrots, walnuts, 1/2 the chicken, onion, eggs, remaining chicken, remaining beets, topping each addition with 1 tablespoons mayonnaise except for the walnuts, onion, and last beet layer.

Remove the glass and cover salad with the remaining 6 tablespoons mayonnaise. Garnish the top with pomegranate seeds.