Forget wooden handle ones. They shed bristles and can be dangerous. (ask me how I know). Get yourself one of these and replace it in a timely manner as it wears
https://www.amazon.com/dp/B00KYSNZS2/?tag=tvwb-20
I have never had one of these brushes lose a bristle..................ever. I have had Weber brushes, Charbroil, and others lose them and actually nearly swallowed one! Because I needed a new brush and was at Walmart instead of Menard's where I usually buy the Libman brushes.
The way the bristles are bonded into the molded plastic head prevents them from coming out. Rather than bundles being pressed into wood or plastic heads. The Libman ones have the bundles literally molded into the molten plastic head and they cannot/do not come out
As an added measure I also will use a bundle of wet paper towels to wipe the grates down after and blast off any remaining deposits