Put the asparagus in a skillet large enough to hold them in one layer (10 or 12 inches diameter). Add 1 teaspoon salt and the water.
Bring to a simmer, then partially cover and cook 10 minutes or so, stirring the spears around now and then, until the they're are tender when pierced but still bright green in color. Drain, arrange on a serving plate and drizzle with the olive oil.
In the same skillet, add water to almost fill the pan. Add the vinegar and bring to a simmer. Tip the eggs into the simmering water and cook just until the whites are opaque and set and the yolks are still jiggly.
Lift the eggs from the water and blot on a towel. Drizzle the asparagus with the butter and cover with the cheese. Place an egg over each portion, season with black pepper and serve.
You can cook the asparagus 1-2 days ahead. Serve warm or cold with the vinaigrette and warm poached eggs.
If you like, you can also poach the eggs 1-2 days ahead. Keep them in the refrigerator in small covered dish, covered with cold water. When ready to serve, carefully reheat them for a few minutes in the simmering water.