To prepare the artichokes: Fill a large pot with water and add the 2 tablespoons salt and the lemon halves. Bring to a boil over medium-high heat. Add the artichokes, cover, and cook for 35 minutes. A large leaf should come off when gently pulled. Set aside and let cool slightly. When cool enough to handle, halve the artichokes down the middle lengthwise. With a spoon or melon baller, carefully scoop out the choke and prickly thorns. Drizzle oil on both the insides and outside leaves of the artichokes. Season with salt and pepper.
To prepare the sausage stuffing: Combine the bread crumbs, garlic, mint, cheese, cooked onion and sausage in a large mixing bowl. Drizzle with oil to moisten until the stuffing binds together; season with salt and pepper.
Preheat an outdoor grill or stovetop grill pan. Preheat the oven to 350 degrees F.
Place the artichokes, cut side down, on the hot grill. Cook for 3 minutes and then turn gently with tongs. Grill the other side for 2 minutes. Spoon the stuffing into the cavities of the artichokes. Drizzle with a little more oil. Place the artichokes in the oven and bake for 10 to 15 minutes until lightly browned. In the meantime, make the aioli.
To prepare the aioli: In a blender, combine the egg yolks, garlic, capers, red pepper flakes and lemon; season with salt and pepper. Blend on medium-high speed and very slowly pour the oil through the pouring hole. If the aioli is too thick, add water. Refrigerate while grilling.
To serve: Put the stuffed artichokes on a platter with the aioli on the side for dipping.