Cook the potatoes with their peel until done, then peel them and let them cool. Cook the balsamic vinegar with honey in a saucepan for a few minutes. Roughly chop the basil leaves.
Heat the oil in a pan and fry the potatoes in it over high heat until golden brown and season with salt. Pour in the balsamic vinegar mixture and toss the potatoes in it.
Mix in the olive oil and basil before serving. Sprinkle with parmesan cheese as desired.