- In a saucepan over medium heat, soften the onion and garlic in the butter. Add the broth and the potatoes. Bring to a boil, reduce the heat and simmer until the potatoes are tender, about 15 minutes. Set aside 18 asparagus tips. Add the remaining asparagus to the saucepan and continue cooking until tender, about 5 minutes.
- Transfer the soup to a blender. Add the grated cheese and purée until smooth. Strain if desired. Add the cream. Season with salt and pepper. Keep warm.
- In a small saucepan of boiling salted water, cook the asparagus tips until tender, about 2 minutes. Remove the tips from the hot water, leaving the water in the saucepan, and plunge them into ice water. Drain and set aside.
- In the same saucepan, add the vinegar to the water, which should be barely simmering.
- Break 3 eggs into 3 small cups or saucers. Slide the eggs into the water and poach until done to your liking, about 3 minutes (the yolks should still be runny). Transfer the poached eggs to paper towels. Repeat with the 3 remaining eggs.
- Ladle the soup into 6 shallow bowls. Place 1 egg on each slice of toast. Gently set 1 egg-topped toast in each serving of soup. Reheat the asparagus tips in the microwave. Garnish the soup with the tips and Parmigiano-Reggiano shavings.