Soak mushrooms in lukewarm water for 30 minutes; cut the chicken in 8 pieces; coarsely chop and place the odori (onion, celery, parsley, carrot, garlic, and rosemary) on a piece of cheesecloth, tie into a bag, and set aside.

Heat the butter and olive oil in a flameproof casserole. When the butter is melted and hot, add the chicken pieces and sauté lightly over medium heat for 10 to 12 minutes. When the chicken pieces are golden, transfer them from the casserole onto a serving dish.

Leaving the casserole on the heat, add the flour to the remaining oil and cook, stirring very well, until golden (about 3 to 4 minutes). Meanwhile, heat the broth to boiling in a saucepan, then pour 1-1/2 cups of it into the casserole and mix very well, to prevent lumps from forming. Meanwhile, drain the mushrooms and clean them, removing all the sand attached to the stems. If the soaking water is saved for another dish, strain it through several layers of paper towels to remove sand. The water may be frozen in ice cube trays.

Put the chicken pieces back into the casserole and add the cheesecloth bag containing the odori. Cover and simmer very slowly for 20 minutes, then remove the cheesecloth bag and taste for salt and pepper. Add the mushrooms and more hot broth if needed, cover, and simmer for 10 to 12 minutes more, stirring every so often with a wooden spoon.

When the cooking time is almost up, beat the egg yolks in a bowl. Remove the casserole from the heat and quickly add the beaten eggs, stirring continuously with a wooden spoon. When the eggs are well amalgamated, squeeze in the juice from the lemon and mix thoroughly.

Transfer to a serving dish and serve hot.