Heat 2 T olive oil in a 12-inch skillet over medium high heat. Add onions and thyme and cook until soft, about 4 minutes. Transfer to a large bowl and reserve. Wipe skillet clean.
Place chickpeas and rice in a food processor, along with garlic, tahini, and zest. Pulse until chickpeas and rice are chopped, but not yet a paste, about 8 pulses or about 10 seconds. Transfer mixture to the bowl with the onions. Add parsley and egg, season with salt and pepper, and mix well to combine. Divide mixture into six 4-ounce patties and refrigerate for 20 minutes.
Heat 2 T of oil in skillet over medium high heat and cook burgers in batches of three, flipping once halfway through until toasted on each side and cooked through, about 2 minutes per side. Repeat with remaining oil and burgers.
Place the burger on your choice of bread, add some of the Cava tzatziki, crumbled feta, baby arugula, and tomatoes. A kiss of fresh lemon to finish the dish is all you need. Enjoy!