Add the chickpeas to a large bowl and use clean hands (get in there!) to mash most of them up . Add the rest of the ingredients to the bowl and use your hands again to mix and mash thoroughly. Use a heaping ⅓ cup to form into 4 compact patties (about ½-inch thick). If patties aren't easy to form it's probably because you need to mash up the chickpeas some more.
In a large skillet over medium heat, add enough oil to generously coat the pan (about 2 tablespoons). Place the patties in the pan (you should hear a sizzle), press down gently and cook for about 5 minutes, or until golden brown and crispy. Flip, gently press again, and cook for another 3 minutes, or until golden brown. Transfer to a plate or sheet pan lined with paper towel.
To make the yogurt feta sauce, in a high-speed blender, add the feta, yogurt, olive oil, and lemon. Blend until smooth (it’s okay if it’s a little chunky). You want the sauce to be thick, not super runny (this may depend on how thick your Greek yogurt is). Taste and season with salt, if needed.
To make the tahini sauce, whisk together all of the ingredients in a small bowl until smooth. If too thick, add water to thin out. The sauce will thicken up in the fridge, so just add water until it's to your desired thickness. Always taste and add more salt, if needed.
If making a burger, toast your buns, add arugula, chickpea beet patty, and top with yogurt feta sauce or tahini sauce (keep extra nearby, you’ll want more!).