Preheat the broiler.

Brush the mushroom caps on both sides with olive oil and sprinkle with ⅛ teaspoon salt, then place face down on a broiler pan. Cook in the oven about 6 inches (15 cm) under the broiler until tender, turning once, about 7 minutes. Cover and put in a warm place until ready to use.

In a medium saucepan, bring at least 2 inches (5 cm) of water to a boil over high heat, then lower the heat to simmer. Add the white vinegar to the water, letting it swirl. Gently break the egg onto a large metal spoon or into a ramekin and add it into the middle of the vortex. Keep stirring so the egg does not stick to the bottom, but keeps turning. Cook for 3 minutes, then remove with a slotted spoon, place on a plate and set aside. Repeat with the second egg.

Mix the egg yolks and lemon juice together with an electric or manual whisk in a stainless steel bowl until the mixture is starting to thicken and is almost double in volume, about 1 minute.

Place the bowl over a saucepan containing simmering water (or in a double boiler), on low heat. Continue to whisk rapidly, being careful not to let the eggs get too hot or they will scramble. Slowly pour in the melted butter and continue to whisk until the sauce is thickened. Add ¼ teaspoon of salt and fresh pepper and whisk until incorporated.

Assemble the benedicts on individual plates by placing the roasted mushroom on the bottom, folding a slice of prosciutto on top, then placing a poached egg on each. Complete with a generous tablespoon (15 ml) of the sauce.

Serve immediately.