Slice the onions into uniform thickness slices. You can choose very thin or medium, which is about ¼-inch.
Add the butter and oil to a skillet. Top with the onions and season them with the salt and pepper.
Cook on medium-low, stirring often until the onions soften. Pour in the molasses.
Continue cooking and stirring every few minutes until the onions are very soft and a deep golden brown.
Deglaze the skillet with vinegar, water or beef broth and scrape the browned bits from the bottom. Let the liquid cook off completely. Cool.
For the Grilled Cheese
Spread mayo on both sides of the bread. Toast one side in the skillet. Flip and build the sandwiches in the pan.
Layer the cheese on the toasted side of one piece of bread. Top with a couple of tablespoons of caramelized onions. Top with another piece of bread, toasted side down.
Notes
The onions make a big batch. If you are only making a few sandwiches, store the onions in airtight containers in the freezer, in single-serve portions for easy meal prep.