Heat the oil in a fondue pot, briefly sauté the garlic. Add the tomato juice and tomato puree, bring to the boil, then remove the pot from the heat. Add the cheese to the hot tomato sauce. Melt over a low heat while stirring, add the thyme.
Combine the cornflour and wine, add while stirring to the fondue, bring to the boil, add the grappa, season, serve immediately.
Serve the fondue with potatoes and bread.