Heat butter and olive oil on medium heat on the stovetop in a large frying pan until sizzling.
Add onions and stir well to coat.
Cook for 2-3 minutes, then stir.
Allow the onions on the bottom of the pan to brown well, but not burn.
Then stir again, scraping up all the dark bits formed on the bottom of the pan each time.
When onions become dry, add a little bit of water - several tablespoons at a time.
Keep doing this, for 30 - 45 minutes, or until you have deep, dark caramelized onions.
Salt onions to taste.
Optional: add a pinch of sugar to sweeten if needed, or a splash of balsamic vinegar for extra flavor.