Position a rack in the center of the oven and heat the oven to 350°F Fill a 12-inch-wide, 2- to 3-inch-deep pan with 1-1/2 inches of water and 1 teaspoons cider vinegar. Bring the water to a simmer over high heat; reduce the heat as necessary to maintain a gentle simmer.
Meanwhile, cook the bacon in a 10-inch skillet over medium heat, stirring often, until browned and crisp, about 7 minutes.
While the bacon is cooking, spread one side of the bread triangles with 1 Tbs. of the butter and arrange butter side up on a small baking sheet. Toast in the oven until crisp and golden, about 10 minutes.
With a slotted spoon, transfer the bacon to a paper-towel-lined plate to drain. Pour off all but 1 Tbs. bacon fat from the skillet. Add the remaining 1 Tbs. butter and the shallot to the skillet and cook over medium heat until softened, about 2 minutes. Add 1/3 cup water and the Brussels sprouts, cream, the remaining 2 Tbs. cider vinegar, and 1/4 teaspoons each salt and pepper; cook, stirring, for 1 minute. Cover the pan and cook, stirring occasionally, until the Brussels sprouts are just tender, 5 to 7 minutes.
While the Brussels sprouts cook, crack the eggs one at a time into a small bowl or teacup and then gently slide each egg into the simmering vinegar water. Poach until the whites are completely opaque but the yolks are still soft, 3 to 4 minutes. With a slotted spoon, remove the eggs from the water (in the same order they were added) and gently blot dry with a paper towel.
Put 2 toast triangles on each of 4 plates and top each toast with an egg. Season the eggs lightly with salt and pepper and spoon the Brussels sprout mixture on top. Sprinkle with the bacon and a few shavings of Parmigiano-Reggiano. Serve immediately.