In a large skillet, heat the vegetable oil over medium heat. Add the thinly sliced onions and cook, stirring occasionally, until they are golden brown and caramelized, about 10-15 minutes. Remove from heat and set aside.
Preheat your grill or barbecue to medium-high heat. Cook the sausages, turning occasionally, for about 10-12 minutes, or until they are cooked through and nicely browned.
If using hot dog buns, lightly toast them on the grill for a minute or two until they are warm and slightly crispy. If using slices of white bread, you can serve them as is or lightly toast them on the grill.
Place a cooked sausage in each hot dog bun or on a slice of bread. Top with the caramelized onions.
Drizzle with tomato sauce (ketchup) and mustard. If desired, you can also add BBQ sauce and/or a slice of cheese.
Enjoy your meal!