Place cauliflower and potatoes in a large saucepan and cover with cold water. Bring to the boil over high heat. Reduce heat to medium and cook until very tender when pierced with a fork (approximately 20 minutes however depends on the size of the potatoes), then drain.
Place the cooked potatoes and cauliflower into a potato ricer. Mash into a heatproof bowl until smooth (we recommend mashing twice). Set aside to cool.
Season with salt and pepper to taste. Add the grated Parmigiano Reggiano (or Grana Padano) cheese and stir with a wooden spoon. Add the eggs, breadcrumbs and parsley, then stir until combined.
Cover the bowl with cling film and leave to rest in the fridge for 20 minutes. This allows the breadcrumbs to absorb any liquids from the cauliflower and potato mixture.
Remove the cauliflower and potato mixture from the fridge. Sprinkle a shallow tray lightly with flour. Roll a tablespoonful of the cauliflower and potato mixture into a ball. Then gently flatten so that it is slightly oval in shape. Repeat with the remaining balls and lightly toss in flour to give a light dusting over the top.
Heat enough sesame oil to cover the base of a large non-stick frying pan. Cook the polpette over medium heat in batches, turning occasionally, for 2 minutes on each side or until golden and cooked through. Add extra sesame oil when the oil has been absorbed from the pan.
Transfer cooked Sicilian-style cauliflower and potato polpette (balls) to a plate lined with paper towel. Serve immediately – buon appetito.