In a high-speed blender, add the feta, yogurt, olive oil, lemon, and garlic. 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.
Heat up the grill on high heat for at least 15 minutes.
In a large bowl, add the beef, beet, soy sauce, garlic powder, cumin, salt, and pepper. Use your hands to mix until just combined. Form into 4 large patties or 6 smaller ones, about ¾-inch thick.
Brush each patty with a bit of oil (this helps prevent sticking) and transfer to the grill for 3-5 minutes, or until there are nice grill marks. Flip and cook for 5-7 minutes more, or until cooked to your liking.
If making a burger, toast your buns, add arugula, beef patty, and top with yogurt feta sauce (keep extra nearby, you’ll want more!).