To prepare tzatziki, place cucumber on paper towels, and squeeze until barely moist. Place in a medium bowl. Spoon yogurt onto several layers of heavy-duty paper towels; spread to 1/2-inch thickness. Cover with additional paper towels, and let stand for 5 minutes. Scrape into bowl using a rubber spatula. Add the salt, crushed garlic, green onions, and extra-virgin olive oil; chill.
To prepare patties, heat 2 teaspoons oil in a medium saucepan over medium-high heat. Add onion; sauté 2 minutes or until tender. Stir in oregano, red pepper, and garlic; cook for 30 seconds, stirring constantly. Stir in broth, lentils, and sun-dried tomatoes; bring to a boil. Cover, reduce heat, and simmer 35 minutes or until lentils are tender; drain. Cool.
Place lentil mixture, cheese, and the next 6 ingredients (cheese through black pepper) in a food processor; pulse until coarsely ground. Divide lentil mixture into 8 equal portions, shaping each into a 1/2-inch-thick patty.
Heat a large nonstick skillet coated with cooking spray over medium-high heat. Add patties, and cook for 3 minutes. Turn patties over, and cook over medium heat 3 minutes. Spread 1 tablespoon tzatziki evenly on each bun top and bottom. Arrange the arugula, patties, and tomato slices over bottom halves of buns; top with remaining bun halves.
Note: You can freeze any uncooked lentil patties for up to 1 month: Separate the patties with wax paper, place in a heavy-duty zip-top plastic bag, remove excess air, seal, and freeze. Thaw in refrigerator before cooking.