Sauce – How to prevent the Béarnaise sauce from thickening over night

A couple things:

• make sure you're removing enough of the fat from the pan. Too much fat will keep the sauce thin (and will also tend to separate out afterwards). Also, fat will not evaporate—if your remaining liquid is fat, the temperature will rapidly rise, and very quickly things will burn.
• depending on what you're adding, you may need to use a thickener. There is a reason three of the five mother sauces have roux. (Hollandaise has egg, which thickens, Tomate has tomato which has lots of pectin, which also thickens, so those two don't need it). A good stock or broth has plenty of gelatin, which will thicken especially upon cooling.
• When you add your deglazing liquid, you need to thoroughly scrape up the fond and stir it in.
• Keep stirring, especially towards the end when its somewhat thick. Also especially if your pan has hot spots.
• Turn down the temperature. This should go without saying when things are burning. Some heat helps with deglazing, but after that you don't actually need anything more than a low simmer — higher evaporates faster (important, since the meat is often resting & cooling) but not burning the sauce is more important.

I'd just simmer and thus reduce it. No added thickener required.