Bring vinegar, granulated sugar, and 1/2 teaspoon salt to a simmer in a small saucepan. Stir in onion and transfer to a small bowl. Let stand, stirring occasionally.
Melt butter in a small heavy saucepan over medium heat. Whisk in flour and dry mustard and cook, whisking, 2 minutes. Add beer, water, brown sugar, and Worcestershire sauce and simmer, whisking, until just thickened.
Fry eggs and toast bread. Reheat sauce, then remove from heat and gradually whisk in cheese until smooth. Season with salt and pepper.
Top each slice of toast with cheese sauce and 2 fried eggs. Toss watercress with oil, a pinch of salt, and drained pickled onions and arrange on top.
- 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