How to Make It
Preheat the broiler. Blanch the asparagus in a large, deep skillet of boiling salted water until crisp-tender, 3 minutes. Using a slotted spoon, transfer the asparagus to a paper towel–lined plate and pat dry; reserve the liquid in the skillet. Arrange the asparagus in 4 individual shallow baking dishes. Brush with the butter and season with salt and pepper. Sprinkle with the cheese and broil 6 inches from the heat until golden, about 1 minute.
Add the vinegar to the asparagus blanching liquid and return to a simmer. Crack the eggs into the simmering water, leaving plenty of space between them, and poach until the whites are set but the yolks are runny, about 3 minutes. Using a slotted spoon, carefully remove each egg from the water, blot thoroughly dry with a paper towel and set on the asparagus. Serve with toast.