Preheat the oven to 400 degrees F.
In a food processor, pulse the garlic and zucchini until roughly chopped. Add the beans, lemon zest, egg, salt, red pepper, flour and basil. Puree until coarsely combined.
Transfer the burger mixture to the fridge to chill, uncovered, until firmer (10 minutes). You can either leave it in the food processor bowl or scrape into a mixing bowl.
Heat a thin layer of coconut oil in a large cast iron skillet.
Using an ice cream scoop (or a 1/2 cup measure), portion the burger mixture and add the balls directly into the skillet (it will be too wet to use your hands). Allow the burgers to get crusty on the first side before disturbing them, about 3 minutes. Using a flat spatula, carefully flip the burgers onto the second side. They should naturally flatten into more of a patty shape, but feel free to gently shape them with your spatula. Cook for another 2 minutes, until nicely browned on the second side.
Transfer the burger to a parchment-lined baking sheet and finish in the oven for 10 minutes, or until the burgers are firm and have developed a falafel-esque crust around the sides.
Meanwhile, make the aioli: in a small bowl, whisk together the mayonnaise, lemon juice, garlic, basil and olives. Alternatively, if you have cleaned the food processor bowl, you can whiz these ingredients in there (no mincing of the garlic, basil and olives necessary).
To serve, slather each bun with the aioli and top with a burger patty. Divided the avocado and sprouts between the sandwiches, and serve immediately.