1. Bring 6 cups water to a boil in a medium saucepan; season with 2 teaspoons salt. Add asparagus tips and blanch until bright green and just tender, 2 to 3 minutes. Transfer to an ice bath with a slotted spoon; reserve for topping. Add trimmings, 1/2 shallot and 1/2 clove garlic to the same pot and return to a boil; reduce heat and simmer the asparagus stock 20 to 25 minutes to allow flavors to meld.

2. In another saucepan, heat olive oil over medium. Add sliced shallot and chopped garlic and cook until softened but not browned, about 5 minutes. Stir in potato and enough of the asparagus stock to cover the vegetables (about 1 cup), and simmer until the potato is tender, 10 to 12 minutes. Strain remaining asparagus stock directly into saucepan with the potatoes. Add remaining asparagus and cook until tender and bright green, about 5 minutes. Remove from heat and let cool slightly. Puree in a blender until completely smooth; remove 1 cup and reserve. Slowly stir in cream, and season to taste with salt and pepper; thin with reserved liquid as needed. Serve hot or chilled, topped with asparagus tips and a drizzle of oil.