Mix ground beef and egg together in a large bowl; add oatmeal, ketchup, mustard, salt, and pepper. Mix well.
Arrange hot dog buns on a baking sheet. Spoon ground beef mixture into each bun.
Bake in the preheated oven for 10 minutes. Add an American cheese slice to each 'boat burger'; continue baking until ground beef is fully cooked and buns are crispy, 10 to 12 more minutes. An instant-read thermometer inserted into the center should read at least 160 degrees F (70 degrees C).