That would be a good idea. I'm not sure I would chop them small but would chop them off and maybe turn them in.

It should work just fine. I do not have enough ready made compost this year, so I have been taking all my veggie food waste through an old food processor I keep in the basement and I pour it all over the garden. I pull the mulch back, pour it and put the mulch back to keep fruit flies away. It works great.

Also, to accelerate the compost, I chop composting material with the lawnmower. It's a lot more work and I can't say I would be doing it every year, but it works.