To prepare aioli, combine first 3 ingredients; cover and refrigerate.
To prepare burgers, combine water, lentils, and 1/4 teaspoon salt in a medium saucepan; bring to a boil. Cover, reduce heat to medium-low, and simmer 20 minutes. Drain; set aside.
Heat a large nonstick skillet coated with cooking spray over medium-high heat. Add the onion, carrot, and 3 garlic cloves; sauté for 3 minutes. Add 1/2 teaspoon salt, mushrooms, marjoram, and pepper; cook 3 minutes, stirring occasionally. Add wine; cook 1 minute or until liquid almost evaporates. Place onion mixture in a large bowl; let stand 5 minutes. Add the lentils, breadcrumbs, 1 tablespoon lemon juice, and egg whites. Cover and chill 30 minutes (to help firm up the mixture).
Divide the lentil mixture into 6 equal portions, shaping each portion into a 1/2-inch-thick patty. Heat the vegetable oil in a nonstick skillet over medium heat. Add lentil patties, and cook for 5 minutes on each side.
Line the bottom half of each hamburger bun with an arugula or lettuce leaf, and top each bun half with a burger, 2 teaspoons aioli, 1 tomato slice, 1 onion slice, and the top half of the bun.