Place rolls, cut sides up, on a baking sheet; coat with cooking spray. Broil 2 minutes or until toasted. Set aside.
Reduce oven temperature to 400; place a baking sheet in oven.
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.
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.
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.