How can I remove large pieces of cured resin (e.g. on a piece that got knocked out of level)?

Depending on the shape and slope of your piece, a sander or hand planer would work to pare down cured resin. A second coat would bring back the gloss and hide the white rough surface left by the planer or sander. You could also try simply applying a second coat from the outset to level a slanted surface.