Make the marinade: mix together the oil, salt, pepper, and the garlic in a small bowl. Put the steak in shallow bowl or a dish and rub the mixture all over the meat. Marinate at room temperature for 1 to 2 hours or, better still, covered in the refrigerator for up to 2 days. Turn the meat from time to time to coat both sides evenly.
To prepare: remove the steak from the refrigerator at least an hour before cooking. When you are ready to grill, prepare a charcoal fire so that you have areas of higher and lower heat.
Drain off any excess oil from the steak. Sear the steak over the high-heat area of the grill for 3 to 4 minutes per side. Remove the grill rack and sprinkle 2 cups soaked hickory or oak chips on the low-heat part of the fire. Replace the grill and place the steak over the low-heat area. Cover the grill, but turn the steak frequently; this should take 10 to 20 minutes more, depending on the thickness of the steak and the heat of the fire. Check the interior of the meat with an instant-read thermometer, and remove the steaks at 5 to 7 degrees below the desired degree of doneness. Let the steak rest, covered loosely with foil, for 10 minutes or so before carving.
To carve, cut the loin meat and the fillet meat away from the bone. Cut into 1/2-inch-thick slices. Arrange meat on a platter with the lemon wedges. Serve with vinaigrette.
Place the dried mushrooms in a glass measuring cup or bowl. Add enough boiling water to cover and soak for at least 30 minutes.
Remove the soaked mushrooms with a slotted spoon. Chop them and set them aside. Carefully pour the soaking liquid into a saucepan, leaving the last bit of liquid and any grit behind. Add the vermouth or sherry and the chicken or veal stock and bring to a boil. Continue to boil until the flavors are concentrated and the sauce is reduced to about 1/2 cup. Set aside while you cook the mushrooms.
Heat the 2 tablespoons olive oil in a large skillet over medium-high heat. Add the fresh mushrooms and shallots or garlic and cook and stir for 2 to 3 minutes. Put in the reserved dried mushrooms and continue to cook for 2 to 3 minutes more. If any liquid accumulates in the skillet, pour it into the saucepan with the reduced sauce. Boil the sauce briefly, if you wish, to intensify the flavors further.
Pour the reduced sauce over the mushrooms and add both vinegars. Using a whisk, beat in enough of the remaining oil to form a slightly thick, homogenous vinaigrette. Taste for salt and pepper and red wine vinegar and serve with steak.