In a large nonstick skillet, heat the olive oil over medium-high heat. When it shimmers, add the onion and cook, stirring
occasionally, for about 5 minutes, until soft and lightly browned. Add the garlic and beets, and cook for about 5 minutes more to soften the raw beets, stirring often. Transfer to a food processor; set the skillet aside (no need to wash it).
Add the brown rice, bread crumbs, salt, and pepper to the food processor. Pulse until the ingredients just come together, about 20 pulses. Adjust the seasoning to taste. Form the mixture into patties with the palms of your hands.
Lightly grease the same skillet you used for the onion, and heat over medium-high heat. When it shimmers, working in batches, add the patties and cook for about 4 minutes on each side, until they are nicely browned. Add more oil to the pan as needed between batches.
Make the pesto: In a blender, combine the cashews, garlic, lemon juice, salt, oil, and water and blend for about 2 minutes, until very smooth. Add the basil and blend again until well combined.
To assemble, on each bun, layer the pesto, a burger patty, arugula, pickled onions, and more pesto.