Trim as much fat as you conveniently can off the lamb shanks, unless they're already trimmed. Don't get too zealous about this; there should be a little fat still there.
In a large skillet pan set over a stove burner on medium-high heat, heat the olive oil until it shimmers. Season the shanks with salt and pepper. Brown the shanks on all sides in the skillet. Remove the pan from the heat.
Place the shanks into a roasting pan or large skillet with a cover. It should be big enough to fit all the shanks without overlapping, but without a tremendous amount of space between them, either. Add all the other ingredients around it except for the wine.
Put the skillet back onto the burner on medium heat. Add the wine. Bring the pan to a boil while whisking to dissolve the pan juices and browned bits. Hold at a light boil for about two minutes, then pour over the shanks.
Add enough water to come about one-third up the sides of the shanks. Cover the roasting pan and put it into a preheated 350-degree oven for 90 minutes. Every 30 minutes, turn the shanks.
After 90 minutes, remove the cover but leave the pan in the oven. Turn the shanks and roast for another 30 minutes. By this time, the meat should be falling from the bones with just a touch of a fork. If not, add more water (if necessary) and continue to cook uncovered until done. Remove the shanks and keep warm.
Strain the pan juices into a tall glass or cup. Discard the vegetables. Let the pan juices settle, and skim off all the fat. If you have more than about a cup, put the liquid into a small saucepan and reduce. Adjust seasonings with salt and pepper.