Stout and Cheddar Rarebit with Fried Eggs

Name: Stout and Cheddar Rarebit with Fried Eggs
Ingredients: 1/3 cup cider vinegar; 2 teaspoons granulated sugar; 1 small red onion, thinly sliced; 1 1/2 tablespoons unsalted butter; 1 1/2 tablespoons all-purpose flour; 1/2 teaspoon dry mustard; 1/2 cup Guinness Extra Stout (without foam); 1/3 cup water; 2 teaspoon packed brown sugar; 1/2 teaspoon Worcestershire sauce; 8 large eggs; 4 thick slices peasant bread (preferably sourdough); 1/2 pound extra-sharp white Cheddar, grated; 1 bunch watercress, coarse stems discarded; 1/2 tablespoon extra-virgin olive oil