
Trim the root and green ends off the leek and slice it in half lengthwise. Place each half with the flat side down and slice into thin half moons. Place the leek pieces in a colander and rinse well.
Place a 5 to 6 quart pot or dutch oven over medium heat and drizzle in the olive oil. Add the leeks and cook, stirring often, until they turn translucent and begin to brown; 3 to 5 minutes. Add the garlic and a large pinch of salt and cook for 2 minutes more, or until the leeks are lightly browned. Some browning will occur on the bottom of the pot — this will create a nice depth of flavor!
Add the broth and parmesan rind, if using. Bring the soup to a boil then reduce the heat to medium low. Cover and let simmer for 10 minutes.
When the pasta is still very al dente, add the asparagus and zucchini. Cover and cook for another 2 minutes. Add the peas, spinach, pesto, and lemon juice and give the soup a few stirs to wilt the spinach. Taste the broth and adjust the salt if necessary. Serve immediately topped with grated parmesan cheese and an extra drizzle of pesto, if desired.