1. Melt the butter in a large heavy saucepan over low heat.

2. Meanwhile, equip a food processor with the slicing disk and thin-slice the carrots by pushing them down the feed tube pulsing all the while. Add to the saucepan.

3. Thin-slice the leeks the same way and add to the pan. Raise the heat to moderate and sauté the carrots and leeks, stirring now and then, until limp, about 10 minutes.

4. Add the chicken broth, salt, and pepper, and bring to a boil. Adjust the heat so that the mixture bubbles gently, cover, and simmer until the carrots are very soft, about 30 minutes. Pour the soup into a large heatproof bowl, cover, and cool.

5. Remove the slicing disk from the food processor and slip the chopping blade into place (no need to wash the work bowl). Purée the soup mixture in two to three batches by churning for 30 seconds. As each batch is pureed, return it to the saucepan.

6. Add the cream and egg yolks to the processor and pulse 2 to 3 times until smooth (again, no need to wash the work bowl). Stir in the puréed soup. Set over low heat and cook, stirring constantly, until lightly thickened, about 3 minutes. Do not let the soup boil or it may curdle. Taste for salt and pepper and adjust as needed.

7. Ladle into heated soup bowls and serve. Or, if you prefer, chill well and serve cold (not the French way, but delicious nonetheless).

Note:
A little chopped parsley or freshly snipped chives makes a colorful garnish, but it's not necessary and a good country French cook would probably do without.