Heat a skillet over medium heat and add 2 tablespoons of the oil. Add the onions, season with salt and pepper and saute until slightly golden, about 10 minutes. Set aside to cool.
Add the brown rice, beets, walnuts, yellow raisins, granulated garlic and cooled onions to a food processor and pulse about 10 times, until the mixture is ground similar to ground beef. Transfer to a bowl and add the eggs, 1 teaspoon salt and some pepper. Mix by hand until uniform, then tightly form into 4 patties.
Heat a nonstick skillet over medium-high heat and add the remaining 2 tablespoons oil. Season both sides of the patties with salt and pepper. Gently add the patties to the oil and sear on each side until crisp and browned, about 10 minutes a side.
Place the patties on 2 to 3 Bibb lettuce leaves. Dollop a bit of Citrus-Caper Aioli on top and serve.
Mix the mayonnaise, capers, honey mustard and orange zest and juice in a bowl. Adjust the seasoning if necessary with salt and pepper.