Heat oil in a large stockpot over medium-high heat. Add onion and sauté for 2-3 minutes, stirring occasionally. Add carrots, celery, mushrooms and garlic and sauté for 4 minutes, stirring occasionally.
Add vegetable broth, water and chickpeas, and stir to combine. Continue cooking until the soup reaches a low boil. Then reduce heat to medium to maintain a low simmer.
Add orzo and stir to combine. Continue cooking for 7-10 minutes or until orzo is tender, stirring occasionally.
Whisk together the eggs and lemon juice in a mixing bowl until combined. Ladle out 1 cup of the hot soup broth, and while whisking the egg mixture with one hand, slowly drizzle the broth into the eggs until combined. You'll want this process to be very slow to avoid scrambling the eggs. Repeat with 1 more cup of hot broth.
After the orzo is tender, remove the pot from heat. Slowly drizzle the egg mixture into the soup until combined while also stirring the soup. The soup should appear creamy.
Stir in the spinach and basil and stir for 1-2 minutes until the spinach has wilted. Add salt and pepper to taste. Serve warm.