I'm still using the grip that both holds and controls the tools, and supports the work - important as the wood gets thinner and the bird more vulnerable to being broken off its perch.
Note how I hold the tool: the thumb of my supporting hand acts together with the fingers of the blade hand. This is still a two-handed, controlled grip!
Make small, quite gentle cuts, with a slice where possible.
Safety! No part of your hand or fingers should be in line with the direction of cut.