Heat 2 tablespoons of the oil in a large saucepan over medium heat. Add the onions and garlic and cook until onions are translucent, about 3-5 minutes. Add the prosciutto and cook until the meat is slightly crispy and the oil is mostly gone. Add the remaining tablespoon of oil to the pan and stir to coat the prosciutto evenly.
Working from the outside in, place the Brussels sprouts cut side down in the pan. (This is important because the center of your pan is the hottest, so by working from the outside in, each will have relatively equal cooking time versus temperature.) Add the stock and maple syrup to the pan. Reduce heat to low, cover, and simmer for 10 minutes. Remove the lid, increase heat to medium-high, and cook for another 5-10 minutes, moving the sprouts around to cook evenly. Cook until all of the liquid has evaporated and the sprouts are tender.