How can I cook with worcestershire sauce in a pan without leaving an impossible to remove char on the pan?
Worcestershire sauce is fairly sweet, it's the sugar that's blackening. There's nothing you can do about it - if you add it to your pan it's going to char. 

The solution is not to put it in the pan, but put it on afterwards or mix it in with the meat before making patties. You'll get a better result that way as well, you're destroying the flavor of the worcestershire sauce by charring it in the pan like that.