Slice the cabbage finely or let a machine do the work for you. Cut the chestnuts into smaller pieces, peel the apples and cut them into pieces.
Take a bit pan and melt a good size chunk of butter. Add the cabbage and fry for a bit while stirring to make sure all cabbage has a layer of butter around it. Once done add a very small amount of water into the pan. It doesn't need a lot as the cabbage will release moisture as well.
Let the cabbage cook til done to your liking. 20-30 minutes depending on the size of the cabbage. Add apples and chestnuts, combined with the balsamic, lemon and sugar. Let it all cook together for a bit before tasting and adding pepper and salt to your liking.
Notes
You can add the apples sooner, depending on your preference. I like mine to have a bite so I add them late, but if you like yours softer, add them right at the beginning of the cooking process.