1. Preheat broiler to high.
2. Place rolls, cut sides up, on a baking sheet; coat with cooking spray. Broil 2 minutes or until toasted. Set aside.
3. Reduce oven temperature to 400°; place a baking sheet in oven.
4. Combine beet and next 6 ingredients (through rice). Combine mustard, eggs, and egg white. Add 1/4 teaspoon salt and mustard mixture to beet mixture; stir well. Spoon about 2/3 cup rice mixture into a (4-inch) round biscuit cutter; pack mixture down. Remove mold; repeat 5 times to form 6 patties.
5. Heat a large skillet over medium-high heat. Add 1 tablespoon oil; swirl to coat. Carefully add 3 patties to pan; cook 3 minutes. Carefully transfer patties to preheated baking sheet coated with cooking spray, turning patties over. Repeat procedure with remaining 1 tablespoon oil and 3 patties. Return baking sheet to oven; bake patties at 400° for 12 minutes.
6. Combine mayonnaise, juice, cheese, and 1/8 teaspoon salt. Place bottom half of each roll on a plate. Divide mayonnaise mixture among roll bottoms; top each with 1 patty. Arrange 1/4 cup arugula on each patty; top with roll tops.