Place chickpeas in a large bowl. Mash with a potato masher (or fork) until mostly smooth (a few small chunks are fine). Set aside.
Combine quinoa with 1 cup water and ½ teaspoon salt in a small saucepan. Bring to a boil over high heat, then reduce heat to low and cover pan. Cook until tender (about 15 minutes).
Drain quinoa in a fine mesh strainer. Transfer to bowl with chickpeas.
While quinoa cooks, heat 2 teaspoons olive oil in a large nonstick frying pan. Add spinach and cook, stirring constantly, until wilted (about 2 to 3 minutes). Season lightly with salt.
Place spinach in fine mesh strainer. Use back of spoon to press out excess water.
Transfer spinach to cutting board, then finely chop. Set aside.
Wipe out large frying pan. Heat 1 tablespoon olive oil in pan over medium-high heat.
Add onion, broccoli, and white part of green onion. Season lightly with salt. Cook until crisp-tender and starting to brown, stirring frequently (about 5 to 7 minutes).
Reduce heat to medium, then stir in chopped spinach. Mix until well combined, stirring constantly (about 2 to 3 minutes).
Add garlic, then cook just until fragrant, stirring constantly (about 30 seconds to 1 minute).
Remove pan from heat, then stir in ¾ teaspoon kosher salt, plus basil, oregano, and red pepper flakes.
Add veggies to chickpeas, along with flour, egg, edamame, walnuts, and green part of green onion; mix well. Season to taste with additional salt, if needed, then shape into 6 patties.
Heat 1 tablespoon olive oil in large frying pan over medium heat. Add patties and cook until browned on one side. Flip, then top with cheese; cook until browned on second side. (You may need to do this in batches, depending on the size of your pan. Add more oil, if needed.)
Spread top half of buns with pesto mayo. Place lettuce and tomato on bottom half of bun, then top with burger and top half of bun.
Serve burgers hot. Enjoy!
Notes
Make sure to read the info before this recipe card for helpful tips, topping ideas, serving suggestions, and more!