Cut bell peppers in half, lengthwise, and take out the seeds and the membrane out. Set aside.
Heat up a medium cooking pan and add oil. Add diced onion and saute until transparent. Add ground beef, cover and cook for a couple of minutes. Break up all the beef clumps, mix well, and add Worcestershire sauce, ketchup, salt, and pepper. Mix well and cook until beef is almost done. Take the beef out of the pan with a slotted spoon and set side.
Add sliced onions to the same pan where you cooked beef and cook them until transparent. Add beef stock, Worcestershire sauce, ketchup, salt, and pepper. Stir cornstarch and cold beef stock until dissolved and stir the mixture into the gravy. Stir everything together and cook on medium heat for about 10 minutes.
Preheat oven to 375.
Place prepared pepper halves in a baking dish. Divide cooked ground beef among the peppers.
Spoon some gravy and onions over each stuffed pepper. Pour remaining gravy into the baking pan, around the peppers.
Bake peppers for 30 minutes. Add a slice of Pepper Jack cheese to each bell pepper and bake for a few more minutes.