Position a rack in the center of the oven and heat the broiler to high. Lay the bread slices and prosciutto strips on a foil-lined rimmed baking sheet and brush the bread on both sides with 2 Tbs. of the olive oil. Broil until the bread is golden brown on both sides and the prosciutto is lightly crisp, 2 to 4 minutes per side. Put the prosciutto strips in a small bowl (they’ll continue to crisp as they cool), and set the bread slices on three plates.
Put the oregano in a small bowl. Heat 1 Tbs. of the oil in a 10-inch nonstick skillet over medium-low heat. Add the garlic to the pan and cook, stirring occasionally, until it’s sizzling and fragrant but not browned, about 30 seconds. Scrape the garlic and oil into the bowl of oregano, stir to combine, and set aside.
Add the remaining 1 tsp. oil to the pan, swirling to evenly coat. Add the eggs and distribute the garlic-herb mixture evenly on top. Season with salt and pepper, cover, and cook until the yolks’ edges have just begun to set, 2 to 3 minutes. (The eggs should cook gently, so lower the heat if needed.) Separate the eggs with the edge of the spatula, if necessary, and slide each egg onto a slice of the bread. Drizzle any remaining oil in the pan over the toasts, top with the prosciutto crisps, and use a vegetable peeler to shave a few strips of Parmigiano over the toasts.