I like to reduce the red wine separately so I can fortify it and build more flavor each step of the way. And a little trick from my time as a saucier, use this reduction to adjust your sauce later.
Simply combine all ingredients in a small pot and place over medium low heat. Simmer until the wine has reduced by half, so you should have one cup. Strain and reserve
Place the milk and water into a pot with a hefty pinch of salt and bring to a simmer. Once gently simmering slowly add your dry polenta while whisking. Continue whisking until all of the polenta is incorporated and is super smooth. Cover and cook over low heat for 20 minutes stirring every now and then, particularly the sides and bottom. Once the polenta is cooked through add a splash of milk if you want to adjust the consistency then add in your grated cheese and whisk. Taste and adjust your salt.
First if you have the time take 1 tbsp kosher salt per shank and season your lamb shanks, place on a sheet tray and keep in the fridge uncovered overnight. Allowing the salt to penetrate the lamb prior to braising is a much nicer experience. It’s more even seasoning and leaving it uncovered in the fridge will let you get a nicer sear later down the road.
To prep the carrots I just cut them into two inch segments then cut those segments in half lengthwise, giving you a nice big face to get some color on. For the shallots just peel and cut in half lengthwise. Peel and smash the garlic.
preheat a saute pan or a dutch oven over medium high heat and add a few tablespoons of neutral oil. if you did not dry brine or preseason your lamb season it now. Once the oil shimmers and begins to smoke, sear your lamb shanks on all sides. Remove the lamb and move on to the vegetables. Pour off any excess fat if it seems like a lot has rendered from the lamb then lay your carrots cut side down cook until you have nice caramelization. Flip your carrots and add the shallots cut side down and cook once more until they’re caramelized. Add the smashed garlic and cook until golden brown, I like to tilt my pan and pool the oil in a corner and let the garlic shallow fry in it.
Deglaze the pan with 1/4 cup of your red wine reduction from earlier and use a rubber spatula to scrape the bottom and pick up all that wonderful fond. Place the lamb shanks back in the pan and cover with beef stock. They should be almost completely submerged, depending on the size of your cooking vessel you might need the full 2 quarts of beef stock.
To cook this on the stovetop just bring to a simmer and cover with a tight lid. Every 30 minutes or so come and flip the shanks. Cook for 2 1/2 hours or until very tender, you should be able to use a fork to dig into it.
To cook in the oven have your oven preheated to 350f/176 . Cover your lamb shanks with a lid and place in the oven for 2 1/2-3 hours.
Whichever way you decided to cook the lamb, remove the shanks from the braising liquid and strain through a fine mesh sieve. Place the braising liquid back on the stove and reduce, skimming any fat you see, until it coats a spoon. Whisk in your butter and taste. Adjust to taste using the remaining red wine reduction and salt.
assemble and serve