In a 1- to 2-quart pan over high heat, bring 1 cup water and the sugar to a boil. Add vinegar and onions, and stir occasionally until boiling again. Remove from heat and let cool, about 30 minutes.
Meanwhile, spread mayonnaise on toast and set each slice, mayonnaise side up, on a plate. Arrange half the prosciutto, then half the avocado, on each slice of toast.
Drain onions well and mound half on each open-faced sandwich. Sprinkle with salt and pepper. Eat with a knife and fork.