In food processor container, place pistachios and parsley and blend until pulverized. Add beets, chickpeas, chickpea or brown rice flour, onion, garlic, tahini, lemon juice, cumin, salt, and black pepper; blend until mixture is tacky and sticks together when squeezed between your fingers. Don’t purée the mixture. If mixture is too dry, blend in additional tahini. And if too wet, blend in additional flour. Form into 4 patties no larger than 1-inch thick (to ensure an even crust and prevent crumbling), place on parchment-lined baking sheet, and refrigerate for at least 30 minutes
Heat grill to medium and grease grates well. Cook burgers for 3 minutes per side, or until a crispy crust has formed and patties are heated through.
To make sauce, whisk together tahini, dill, garlic powder, salt, lemon zest, and 2 Tbsp warm water until you have as lightly thick spreadable consistency. If needed, add more water, 1 Tbsp at a time, to thin.
Serve burgers topped with tahini sauce.