Spiced green tahini sauce: Pulse garlic, cilantro, parsley, cumin, and 2 teaspoons salt in a food processor until similar in texture to pesto.
Add tahini and lemon juice; process for seconds. With motor running, gradually drizzle in 1⁄2 cup (120 ml) water and process, adding more water to thin if needed, until sauce is light green and the consistency of sour cream. Season with salt.
Falafel: To the bowl of a 2-quart (1.9 liters) food processor, add the chickpeas, onion, parsley, garlic, lemon juice, spices, and eggs. Pulse the ingredients together until just incorporated, forming a wet paste. Don’t over-process.
Transfer the chickpea mixture to a large bowl and add the breadcrumbs, mixing until just incorporated. Cover with plastic wrap and refrigerate for 1–2 hours, or overnight.
Pickled red onions: Place onions in a bowl. Heat vinegar, water, sugar, spices, and salt in a small saucepan over high heat, whisking until sugar is dissolved. Bring to a boil, then strain and pour over onions. Press onions down with a spoon until submerged.
Let rest until onions are softened and liquid has cooled, about 25 minutes.
Cabbage slaw: In a bowl, combine all ingredients.
Remove the chilled falafel mixture from the refrigerator and shape into 6 patties.
In a large nonstick skillet, heat about 1 tablespoon of oil over medium heat.
Cook burgers in batches of 3 for 3–4 minutes each side, flipping halfway through. Once they are golden brown and crispy, transfer them to a paper-towel-lined plate and sprinkle lightly with salt.
Assembly: Spread about 1 tablespoon spiced green tahini sauce on the bottom of each bun. Place a burger on each bun, topping each with a tomato slice, pickled red onions, and cabbage slaw.