Cut tips from 6 asparagus spears, about 1 1/2 inches from the top; reserve for garnish later. Chop the remaining asparagus into chunks.
Heat olive oil in a soup pot over medium heat; cook and stir shallot in the hot oil until softened, about 3 minutes. Add chopped asparagus; season with mint, salt, and pepper. Cook until asparagus are slightly tender, about 3 minutes.
Pour chicken stock into asparagus mixture; bring to a boil. Reduce heat and simmer until asparagus are tender, 12 to 15 minutes. Stir in lemon zest.
Blend soup using a hand blender or a food processor until smooth.
Bring a small pot of lightly salted water to a boil; cook asparagus tips in the boiling water until just turning bright green, about 2 minutes. Strain asparagus tips and run under cold water to stop the cooking process. Garnish soup with asparagus tips and hard-boiled egg.