Distil neutral spirit as vodka, but then redistil through the gin botanicals basket containing only mint to infuse with the neutral spirit.
Distil neutral spirit as vodka then try adding Still Spirits Crème de Menthe peppermint liqueur flavouring (taking care to adjust the dosage as a lower dose may be required).
You could also add fresh or dried mint to clean distilled spirit and leave to soak until the flavour infuses the spirit. You could crush the mint and add directly to the filtered 40% ABV spirit and filter the mint leaves out once the desired flavour is reached.
