Prepare the barbecue to medium heat or if using oven, preheat to 425°F. Whisk vinegar, shallot and sugar in small bowl until sugar dissolves. Gradually whisk in oil. Season dressing with salt and pepper.
Arrange asparagus on rimmed baking sheet. Drizzle with 2 tablespoons dressing. Sprinkle asparagus with salt and pepper. Grill asparagus until crisp-tender, turning occasionally, about 6 minutes, and remove from oven. On the same baking sheet, drizzle the asparagus with additional 1/3 cup dressing; top with prosciutto and sliced cheese. Bake until cheese melts, about 5 minutes.
Place greens in large bowl. Add 1/4 cup dressing and toss to coat and place in salad bowls or plates. Using spatula, divide asparagus among 6 plates on top of the greens. Sprinkle pine nuts over salads and serve with remaining dressing.