Cut small slits in the skin of the duck breasts, making sure the flesh is not damaged. Season with salt and pepper.
Heat a heavy based (preferably iron) frying pan and put the duck breasts into it skin-side down. Sear them for a couple of minutes, then turn the heat down and fry for another 5 minutes. Turn them over and cook for a further 3-5 minutes, depending on how well-done you like your duck. Transfer them to a plate and rest under cover.
Pour the duck fat away (keep this for cooking other dishes) then add the honey and balsamic vinegar. Using a wooden spoon scrape up all the browned bits and reduce the sauce to a thick syrup. This will take about 2-3 minutes.
Return the duck breasts back to the pan, coat them evenly with the glaze by turning them around a few times.
Put the glazed duck breasts onto a plate and slice them diagonally.
Serve them with the remaining syrup - and chopped walnuts if you like.