6 small spring onions, bulbs only, coarsely chopped (about 1 1/2 cups)
3 medium spring garlic bulbs, 1-2 garlic scapes, or 2 regular garlic cloves, thinly sliced
4 cups low-sodium chicken broth
1/2 pound asparagus, sliced on a diagonal 1/2-inch thick
1/4 pound sugar snap peas, sliced on a diagonal 1/4-inch thick
2/3 cup shelled fresh peas (from about 2/3 pound pods)
2 large eggs
1 Tbs grated Parmesan plus more for serving
1/4 cup torn fresh basil leaves
1/4 cup torn fresh mint leaves
1 1/2 tsp (or more) fresh lemon juice
Heat oil in a large heavy pot over medium heat. Add carrots, spring onions, and garlic and season with salt. Reduce heat to medium-low, cover and cook, stirring occasionally, until vegetables are soft, 15-20 minutes.
Add broth and bring to a boil. Add asparagus, sugar snap peas, and peas and cook until vegetables are crisp-tender, about 3 minutes.
Meanwhile, beat eggs in a small bowl with 1 tablespoon Parmesan, a pinch of salt, and 1 tablespoon water.
Reduce heat to low and stir basil and mint into soup. Drizzle in egg mixture in 4 or 5 spots around pot. Let stand for 1 minute so egg can set, then gently stir in 1 1/2 teaspoons lemon juice. Season soup with salt and more lemon juice, if desired. Serve soup topped with more Parmesan.