Debut will be recording what is on your screen, what you can see - so it will capture whatever is within the Debut frame you've set.
If you have space on your screen, you could have 2 browser windows open and capture your target on one of them and watch/do whatever you want on the other, but you'd have to be very careful not to interfere with the recording area.
That's all I can think of, off the top of my head, as they say 😁