Preheat the oven to 350°F.

Get some foil and make a well in the center of it. Strip off some of the outer papery covering on two heads of garlic and then slice through the top fourth of the garlic head on the stalk end, not the root end. Put the garlic in the well you made in the foil, then drizzle 1 tablespoon oil over the open cloves. Close up the foil and roast in the oven for 1 hour and 15 minutes.

After about 45 minutes or so, start the stew. Get a large sauté pan and pour in the remaining 4 tablespoons of olive oil. Get it hot over medium to medium high heat for 2 to 3 minutes. Pat the lamb pieces dry with paper towels, and salt generously. Working in batches, brown the lamb pieces on all sides in the sauté pan. Once the meat has browned, transfer the meat into a large Dutch oven or other heavy pot with a lid.

Meanwhile, remove the roasted garlic and let cool a little then squeeze out the roasted cloves and mash into a little bowl. Lower the oven temperature to 300°F.

Once the lamb pieces have all been browned and removed from the pan, add the chopped shallots, celery, carrot, and the white parts of the chopped green garlic to the pan and stir to combine. Sprinkle everything with salt. Sauté for 3 to 4 minutes, stirring often. When the vegetables begin to brown, transfer them into the Dutch oven with the lamb.

Add the mashed roasted garlic into the sauté pan and let it sizzle a little. Pour in the white wine and turn the heat to high. Deglaze the pan by scraping the bottom of the pan with a wooden spoon to get all the browned bits, and mix the roasted garlic in well. Once this comes to a rolling boil, let it cook down for 2 to 3 minutes, then pour it over the lamb and veggies in the Dutch oven.

Pour enough hot water into the Dutch oven to almost cover the lamb. Cover and cook in the 300°F oven for 2 hours and 15 minutes.

If you are using bone-in lamb pieces, remove the Dutch oven from the oven and use tongs to remove all of the lamb pieces from the stew. Place on a baking sheet, spread out to cool quickly.

There should be a layer of fat on top of the stew. Skim this off with a large metal spoon or use a fat separator.

Once the meat has cooled enough to touch, use your fingers to remove all of the bones from the meat. Discard the bones and return the meat to the stew.

Taste the stew and add more salt if needed. At this point you can make ahead a day or two.

To finish, add the potatoes to the stew and turn the oven up to 350°F. Cover and cook for an hour.

When the potatoes are done, take the stew out of the oven and set it on the stovetop. Pour in the peas and the light green parts of the green garlic and cover for 2 to 3 minutes. Right when you serve, taste the stew again. Add some salt and lemon juice if you want.