Make sure everyone in your household is following the same ground rules as you are. Otherwise, the dog will not learn to stop begging.  If even one person in your household gives in and feeds from the table, your effort will be compromised. Your dog will learn quickly which people will or will not give it food. Help your family members or roommates understand that it is for the dog’s sake that the begging should stop. The dog should have a balanced diet and maintain a healthy weight for a long life, and feeding from the table will compromise those goals. In addition, begging is a habit that undermines the joy of living with a well behaved house pet. If you slip-up even once, that will give your dog motivation to keep begging.  It’s not the end of the world if this happens, but you will need to start again. Remember that consistency is key with any dog training.  “No” means “no” and you need to follow through and not give in to your dog’s wishes. Your dog is not deprived, will not starve, and will not hate you for this later.  Guilt is a human emotion. Your dog will not hold a grudge against you later for not feeding it table scraps. You can treat your dog later with something healthy if it makes you feel better. Use the treat to reinforce an old command or to start teaching a new one. Do not give treats that are not earned. That means that the behavior is your idea, not your dog’s. Your dog should stop begging within a couple of weeks or so, but you (and everyone in the household) need to remain vigilant. If there is no pay off, the dog will eventually stop trying, especially if you start administering time outs. If your dog is more persistent than you are, you may want to get a professional and reputable dog trainer involved with helping break the habit.  Chances are good that this is not the only bad habit that needs to be broken. Both you and your dog may need to revisit obedience classes to brush up on your commands. Contact your local animal shelter or dog-boarding business. They may have a training program themselves, but if not, can probably refer you to one.
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One-sentence summary -- Get everyone involved. Be consistent. Don't let guilt get in the way. Don't give up. Hire a professional.


Wash the shanks and use a sharp knife to remove some of the larger deposits of fat, but don't trim the shanks of all visible fat. The fat will render and contribute to the flavor of the final dish. Pour the oil into a large dutch oven or another oven-safe dish and place it over medium high heat. Allow it to heat completely, until it begins to smoke just slightly. Season the shanks with salt and pepper on all sides. Place them in the hot oil and brown them on all three sides. Brown each side for about 4 minutes, enough to get a good sear on each side.  Don't cook the lamb shanks all the way through. Browning them will bring out the deep flavor of the shanks, but longer cooking will bring about the soft, fall-apart texture you want. Make sure the oil is nice and hot before you put the shanks in the pot. Arrange the vegetables and garlic cloves around the lamb shanks, and drop in the peppercorns. Pour the wine over the entire contents of the pot. Let the red wine come to a boil, and boil it for three minutes. Add the water and reduce the heat to bring everything to a quick simmer.  Boiling the wine for three minutes will lower the alcohol content a bit, leaving the deep flavor of the wine behind. With the addition of the water, the shanks and vegetables should be fully submerged in liquid. If they aren't, add a little more water. If you don't have a tight-fitting lid for the dutch oven, cover it tightly with aluminum foil. Place it in the oven and braise it for 1 hour and 30 minutes. Every half hour, remove it from the oven and turn the shanks so they cook evenly. After 1 hour and 30 minutes, the shanks should be quite tender. If they aren't, return them to the oven and continue braising, checking every 15 minutes until they've reached the correct texture. Transfer the cooked lamb shanks to a serving platter. Pour the braising liquid through a fine-mesh sieve to strain out the vegetables and keep the liquid. Pour the liquid into a saucepan and cook it over medium heat, stirring often, until it has reduced and become a thick sauce.  Season it with salt and pepper to taste. To make the sauce thicker, add a teaspoon of cornstarch. Pour the braising liquid over the lamb shanks and serve the dish with roasted vegetables or mashed potatoes. Each shank is enough for one serving.
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One-sentence summary --
Preheat the oven to 325 °F (163 °C). Wash and trim the shanks. Heat the oil. Brown the lamb shanks. Add the vegetables, peppercorns and wine. Cover the dish and transfer it to the oven for braising. Strain and reduce the braising liquid. Serve the lamb shanks.