Roll each prosciutto slice into a cylinder and cut into 3/4-inch segments; put into a saucepan. Stir onion, tomato, and Pinot Grigio with the prosciutto; cook over medium-high heat until most of the wine has evaporated, about 10 minutes.
Heat olive oil in a small skillet or omelet pan over medium heat. Beat egg and egg whites together in a small bowl; pour into the hot skillet and tilt the skillet around so the egg covers the cooking surface completely. Cook egg undisturbed until no longer wet on top, about 5 minutes.
Pour the prosciutto mixture over one half of the egg; top with Cheddar cheese. Lift the opposing half of the egg with a spatula and fold over the filling. Continue to cook until the cheese melts, about 2 minutes more.