Prep the vegetables: Slice leeks thinly, then rinse well to remove grit. Dice the carrot and potato, slice celery, and cut asparagus into 1-inch pieces.
Mince the garlic.
Sweat the aromatics: Warm 2 tablespoons olive oil in a large pot over medium heat. Add leeks, carrot, and celery with a pinch of salt. Cook 6–8 minutes, stirring, until leeks are soft and glossy but not browned.
Add garlic: Stir in garlic and cook 30 seconds until fragrant.
Keep heat moderate to avoid burning.
Build the base: Add potato (if using), bay leaf, and broth. Bring to a gentle boil, then reduce to a simmer. Cook 8–10 minutes, until potato is just tender.
Add spring vegetables: Stir in asparagus.
Simmer 3–4 minutes until crisp-tender and bright green. Add peas and cook 2–3 minutes more. If using spinach, stir it in now to wilt.
Season and finish: Remove bay leaf.
Stir in lemon juice and herbs. Taste and season with more salt, pepper, and lemon as needed. For a silkier finish, swirl in a teaspoon or two of olive oil or a small knob of butter.
Serve: Ladle into warm bowls.
Finish with extra herbs, a crack of pepper, or a sprinkle of grated Parmesan if you like (skip for dairy-free).