Heat the oil in the caquelon, press and add the garlic, sweat briefly. Add the tomato juice, puree and sugar, bring to the boil, then remove the caquelon from the hob. Coarsely grate the cheese, add to the hot tomato sauce one portion at a time. Melt on a medium heat while stirring, add the thyme.
Combine the cornflour and wine, keep stirring and add to the fondue, bring to the boil, add the grappa, season, serve immediately.
Peel the hot potatoes one portion at a time if desired, cut into quarters, transfer to plates. Pour the fondue over each portion. Keep the rest of the fondue simmering on a low heat at the table on a rechaud. Stir occasionally with a ladle.