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Bring vinegar, granulated sugar, and 1/2 tsp salt to a simmer in a small saucepan. Stir in onion and transfer to a small bowl. Let stand, stirring occasionally.
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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.
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Fry eggs and toast bread. Reheat sauce, then remove from heat and gradually whisk in cheese until smooth. Season with salt and pepper.
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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.