Melt butter in a large heavy bottomed saucepan. Cut the leeks in half lengthwise and slice into small pieces. Wash thoroughly. Dice the potatoes and carrots and add with the leeks into the saucepan. Cook for 5 minutes while stirring occasionally to prevent them 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 the time, stirring constantly. Season with salt and pepper to taste. Sprinkle with fresh parsley. Serve at once