Melt the butter in a large heavy bottomed saucepan. Cut the leeks in half lengthwise and wash thoroughly. Slice. Dice the potatoes and carrots and add with the leeks to the saucepan. Cook five minutes while stirring occasionally to prevent vegetables from browning.
Add flour stirring constantly; remove from heat and add broth all at once while continuing to stir. Reduce heat and let simmer until vegetables are tender.
Beat egg yolks into a soup tureen. Pour soup over egg yolks, a little at a time, stirring constantly. Season with salt and pepper to taste.