Fish stock: Fill medium saucepan with 3 1/2 cups cold water Add fish bones and head. Bring to simmer. Add reserved vegetable trimmings. Cook for 20 minutes to reduce to 2 1/2 cups. Strain.
Preheat oven to 400 degrees. Heat platter for eggplant and small casserole dish for rice. Bring pot of salted water to boil. Add eggplant halves. Cook about 10 minutes, until tender. Drain and set aside on warming platter.
On stovetop, heat a casserole dish, add 1 tablespoon clarified butter and rice. Saute for 30 seconds. Add bay leaf and 2 1/2 cups of the fish stock. Season generously with salt and pepper. Cook uncovered on middle shelf of oven for 20 minutes, or until tender and the rice has absorbed all of the stock.
In a large, hot saute pan, add 1/4 cup clarified butter. Add the pieces of sole fillet and season generously with salt and pepper. Add wine, cover, and simmer over medium heat for about 5 minutes or until just done. Set aside fillets on platter near stove and cover to keep warm. Reserve juices in the pan.
In large pot, heat 1/4 cup clarified butter. Add shallots. Saute for 30 seconds. Add peppers and 1 tablespoon chives. Saute another 30 seconds. Add mushrooms and 1/2 of lemon juice. Lower heat and simmer (as mushrooms extrude their liquid). Add cooked rice to vegetables and mix well. Add the juices from the fish pan. Add 1/4 cup of the cream. Mix well.
Fill eggplants with rice. Gently lay the fish fillets over the rice. Moisten with a little of the reduced veal stock. Cover to keep warm.
Heat oven to broil. In a small saucepan, add 2 tablespoons of veal stock. Bring to a boil, then remove from heat and add the butter and egg yolk. Whisk furiously. Add the remaining lemon juice and the remaining 1/4 cup of cream. Continue beating until smooth and creamy. Pour the sauce over the stuffed eggplants. Dust with chives and place under the broiler to flash heat.
Veal Stock: In a large stockpot, cover the bones with the cold water and bring to a boil. Continuously remove any scum that rises to the surface. Add the vegetables and seasonings. Bring to boil and reduce to simmer. Simmer for about 5 hours or until reduced to 4 cups.
A specialty of the Hotel Meurice, Paris, France