Anybody Glaze Your Burgers?

One of my fav burgers is a Hawaiian… that has cole slaw on the bottom bun, two 2.5oz smashed patties (one topped with swiss, and the other pepperjack), topped with a fresh pineapple ring and teriyaki sauce.

It’s delightful, but the teriyaki seems a bit of an afterthought… and I am looking to get more of its flavor into the beef. I’ve thought about under cooking the beef just a bit, and throwing the patties and pineapple ring into another skillet with teriyaki and kind of caramelize the sauce onto them… and then top with cheese. Think this might work well with BBQ sauce for a western bacon cheeseburger as well.

Any of you do such a thing?