Place first 3 ingredients in a medium saucepan; bring to a boil. Cover, reduce heat, and simmer 30 minutes or until tender; drain. Discard bay leaves. Place lentils in a large bowl; partially mash with a potato masher. Cool slightly.
Heat oil in a medium nonstick skillet over medium-high heat. Add onion and carrot; sauté 5 minutes or until tender. Cool slightly.
Add onion mixture, cheese, and next 8 ingredients (cheese through egg whites) to lentils; stir well to combine. Cover and chill 45 minutes. Divide mixture into 8 equal portions, shaping each into a 1/2-inch-thick patty.
Heat a grill pan coated with cooking spray over medium-high heat. Add half of patties, and cook 5 minutes on each side or until done. Repeat procedure with remaining patties. Spread 1 teaspoon mustard on top half of each bun. Place 1 patty on bottom half of each bun, and top each serving with 1 tomato slice, 1/4 cup arugula, and top half of bun.