Heat the oil or vegan butter in a large Dutch oven over medium-high heat.
Add the chopped onion, celery, and carrots and sauté for 5 to 7 minutes, until the onion is soft and beginning to brown.
Stir in the minced garlic and Italian seasoning and cook for about 30 seconds until fragrant.
Sprinkle the all-purpose flour over the vegetables and cook, stirring, for about 1 minute to remove the raw flour taste.
Add the drained chickpeas and the broth, stir well to dissolve the flour, then cover and bring the soup to a boil over high heat.
Uncover, stir in the dry orzo, reduce heat to medium, and simmer for about 10 minutes, stirring frequently, until the orzo is tender and not sticking to the pot.
While the soup simmers, make the gremolata by combining the finely chopped parsley, lemon zest, and a pinch of salt in a small bowl; set aside.
When the orzo is cooked, reduce heat to low and stir in the baby spinach and lemon juice (start with juice of 1/2 lemon); cook just until the spinach wilts.
Season the soup with salt and black pepper to taste, divide among bowls, and top each serving with the parsley-lemon gremolata.