Prepare rice according to package directions. Cool completely.

In food processor, pulse beans, egg and water until mashed.

Transfer bean mixture to large bowl. Add rice, beets, 1 1/2 ounces goat cheese, dill, salt and pepper; stir until combined. Divide mixture into six portions and shape into 1/2-inch-thick patties.

In large, nonstick skillet over medium heat, heat 1 tablespoon oil. Cook patties in two batches, 3-5 minutes per side, or until golden brown and heated through, adding remaining oil as necessary.

Place two lettuce leaves together, overlapping ends to form pocket. Place patty inside lettuce pocket; top with arugula and remaining goat cheese. Drizzle with Sriracha. Repeat with remaining burgers.