Heat enough oil to cover the bottom of a large skillet over medium-high heat.
Halve the onions and slice thin. Add to skillet and stir to coat in oil. Spread the onions out and let cook for 10 minutes.
Stir in the sugar and a few good pinches of salt.
Let the onions cook for about 30 minutes, making sure they don't get dry (if they do, add a bit more oil). Do not stir them too much, the idea is get them good and browned and soft.
When the onions are just about done add a few splashes of Balsamic vinegar, to taste. Stir.
Cook dogs as desired and lightly toast the buns if you wish.
Place dogs into buns. Divide the onions and cheese between the dogs. If you want the cheese to get good and melted, throw the dogs under the broiler for a minute.