Add 1 tablespoon of butter and olive oil to a large pan over medium heat. When the butter is bubbly and frothing, add all of your sliced onions. 

Add a pinch of salt and stir everything to combine. 

Reduce heat to medium-low and cook onions, stirring often, for 20 minutes. 

After 20 minutes, your onions should be tan-colored, but not browned. You might start to see dark brown spots forming on the bottom of the pan, though, and if you do see that, you can add a tablespoon or two of water to the pan with the onions and use that water to stir the dark parts at the bottom to keep things from burning. 

Continue cooking the onions over medium-low heat for another 10 or 15 minutes. Remember: if you see dark spots or dry areas on the pan, add a tablespoon of water. The water should cook off quickly, but it will help to temper the pan temperature and help the onions to cook through without burning. 

After the onions are nice and brown, remove them from the pan to a bowl to cool for about 20 minutes.

When the onions are cooled, add them to a blender or food processor and pulse until they are very small pieces. If you do not have either of these machines handy, you can dice the onions over and over until they are in tiny pieces. 

Add them to 1 cup of mayonnaise and stir to combine well. Store in a sealed container in the fridge for up to a week.