Preheat a broiler or a lightly oiled stove-top grill pan.

Meanwhile, bring the water to a boil in a medium saucepan. Salt the water, add the vinegar, then lower the heat so the water barely bubbles. One at a time, break the eggs into a custard cup or small bowl, then slide gently into the water. Allow the eggs to cook for 30 seconds. Then, with a slotted spoon, gently lift and shape the whites around the egg yolks. Continue to cook until the whites are just set and the yolks are 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, then place an egg on each toast. Drizzle each toast with 2 teaspoons Basil Oil, then grind a little black pepper over each. Serve immediately.