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Bring the water to a boil in a medium saucepan. Salt the water, add the vinegar, and then lower the heat so the water barely bubbles. One at a time, break the eggs into a custard cup or small bowl, and then pour into the water, trying not to break the yolks. With a slotted spoon, gently lift and shape the whites around the egg yolks until glazed but still liquid, about 2 1/2 minutes longer.
While the eggs poach, toast the bread slices on both sides. Put a slice of toast on each of 2 plates. Immediately top with the prosciutto so that it softens from the heat of the toast. Lift the poached eggs out of the water with a slotted spoon, letting excess water drip off, and then place an egg on each toast. Drizzle each toast with 2 teaspoons chive oil and then grind a little black pepper over each. Serve immediately.