1. Preheat broiler.
2. Heat a large nonstick skillet over medium-high heat. Add 1 teaspoon oil to pan; swirl to coat. Add onion and garlic; sauté 3 minutes or until onion is tender. Remove from heat.
3. Place 1 bun in a food processor; process until coarse crumbs form. Remove breadcrumbs from food processor; set aside.
4. Combine mushrooms, lentils, 2 teaspoons mustard, and the next 3 ingredients (through salt) in food processor; pulse to combine.
5. Combine onion mixture, breadcrumbs, mushroom mixture, and egg in a large bowl, stirring well. Divide mixture into 4 equal portions, gently shaping each into a 1/2-inch-thick patty. Heat a large nonstick skillet over medium-high heat. Add the remaining 2 teaspoons oil to pan; swirl to coat. Add patties; cook for 4 minutes on each side or until browned. Remove patties from pan; arrange in a single layer on a baking sheet. Sprinkle patties evenly with cheese. Place under broiler for 2 minutes or until cheese is softened.
6. Combine canola mayonnaise and remaining 2 teaspoons mustard in a small bowl. Spread the top half of each of the remaining 4 buns with 2 teaspoons mayonnaise mixture. Top each bottom half of bun with 1 patty. Divide arugula evenly among burgers; top with top half of bun, mayonnaise-coated side down.
MyRecipes is working with Let's Move!, the Partnership for a Healthier America, and USDA's MyPlate to give anyone looking for healthier options access to a trove of recipes that will help them create healthy, tasty plates. For more information about creating a healthy plate, visit www.choosemyplate.gov.