In a medium, heavy bottomed stock pot, melt 4 oz of the butter over low heat. Add the leeks and onions. Cook over low heat, stirring occasionally, until the onions are translucent.
Roughly chop 4 of the potatoes and add to the pot. Sauté for 2 minutes.
Add the chicken stock and bring to a strong simmer over medium heat. Allow to simmer until the potatoes are tender, 10–15 minutes.
Cut the remaining potato into 1⁄4 inch dice. Place in a small saucepan with 1 Tbsp salt and enough cold water to cover. Cook over medium heat until the potato is tender, about 10 minutes. Strain and reserve for service.
Puree the chicken stock and rough cut potatoes with a blending stick or in a blender. Return the soup to the stove and place over low heat. Add the heavy cream, remaining 4 oz butter, thyme, and truffle oil. Season with salt and pepper to taste.
At time of service, heat 4 soup bowls. Add the diced potatoes to the soup and divide the soup evenly among the bowls. Garnish each bowl with chopped egg, 4 truffle slices, and chopped chives. Serve immediately.