Mix ground beef, green bell pepper, finely chopped onion, Worcestershire sauce, soy sauce, yellow mustard, garlic, oregano, salt, and pepper together in a large bowl. Form mixture into 4 patties that will fit inside the rolls.

Heat vegetable oil in a large skillet over medium-high heat. Cook patties until browned on the bottom, about 6 minutes. Reduce heat to medium-low and flip; cook for 4 minutes. Top each patty with a slice of Cheddar cheese; cook until cheese starts to melt, about 2 minutes. An instant-read thermometer inserted into the center of each patty should read 150 degrees F (66 degrees C) for medium-well.

Mix ketchup and mayonnaise together in a small bowl. Spread evenly over both sides of each bun. Transfer a patty to each bun; top with sliced cabbage, tomatoes, and onion.