Dice 4 carrots, the onions, and washed leeks. Place in saucepan with the butter. Cover and simmer ½ hour. Add potatoes, water, salt, and pepper. Cover and simmer another hour.
Peel 2 carrots and cut into lengthwise slices 1/16 inch thick. Cut these into 1/16-inch strips and then cut into 1/16-inch cubes, thus cutting them à la brunoise. Melt chunk of butter in small skillet. Add brunoise-cut carrots. Place a piece of buttered foil on top of the carrots and then cover pan with a lid. Let simmer over very low heat for about 20 minutes.
When potage is done, press through food mill. Then pour back into saucepan. Heat. Mix yolks and crème. Add about 1cup hot potage to this liaison, and then pour into potage. Heat but do not boil. Fill soup tureen with potage and garnish with cooked carrot cubes.