Cut each onion through the stem end into quarters, leaving the quarters attached at the root end, and set aside.

To make a bouquet garni, gather together the leek, celery, parsley, thyme and bay leaves and tie securely with kitchen string; set aside.

In a large, heavy sauce pan over high heat, warm the olive oil. Place enough of the short ribs in the pan to cover the entire bottom. Cook, turning once, until well browned on both sides, 2-3 minutes on each side. Transfer to a platter and repeat with the remaining ribs, transferring them to the platter as well.

Reduce the heat to medium and add the onions. Sauté to color lightly on all sides, 4-5 minutes, then transfer to the platter. Add the carrot, celery and garlic and sauté until lightly colored, about 5 minutes.

Drain off any excess grease and add the red wine. When the wine boils, deglaze the pan by stirring to dislodge any browned bits from the pan bottom. Add the tomato and return the onions to the pan. Pour in the beef stock, add the-bouquet garni, salt and peppercorns and return to a boil. Return the short ribs to the pan, reduce the heat to low, cover and cook until the meat falls easily from the bones, 1½-2½ hours.

While the ribs are cooking, place a large frying pan over medium heat.

Add the butter and when it melts, add the mushrooms. Sauté until very tender, 4--5 minutes. Remove from the heat and set aside.

When the meat is ready, using tongs, transfer the ribs to a warmed deep platter. Transfer the onions to the platter as well, arranging them around the meat; keep warm.

Strain the cooking liquid through a fine-mesh sieve into a clean saucepan and place over medium-high heat. Skim off any oil from the surface and reduce the liquid by one-third. Add the mushrooms and any accumulated juices, the parsley and thyme and bring to a boil. Boil for 5 minutes, then pour over the ribs and serve.