Heat 1 tablespoon vegetable oil in a medium saucepan over medium-high heat. Add the onion, carrot, bell pepper and 2 tablespoons ketchup and cook, stirring occasionally, until the vegetables are slightly soft, about 3 minutes. Add the garlic and cook, stirring, 30 seconds. Add the lentils, oregano and 4 cups water; bring to a boil and cook 5 minutes, then reduce the heat to medium low and simmer until the lentils are tender and the water is absorbed, 35 to 40 minutes (add up to 1 more cup water if necessary). Season with salt and pepper.
Heat the remaining 1 tablespoon vegetable oil in a large nonstick skillet over medium-high heat. Add the beef and cook, breaking it up with a spoon, until it begins to brown, about 2 minutes. Add the Worcestershire sauce and the remaining 6 tablespoons ketchup and cook, stirring, until combined. Add the lentil mixture and 1 cup water and cook, stirring occasionally, until the lentils are soft and the mixture thickens, about 8 minutes. Season with salt and pepper.
Fill the buns with the cheese and lentil mixture. Top with pickles and/or pickled jalapenos.