Remove the peel from the potatoes and dice them.
Remove the skin from the onion and the garlic clove and chop finely.
Heat the olive oil in a medium frying pan. Sauté the onion and garlic in it. Add the potatoes, the soup and the martini. Cover and simmer gently for 20-25 minutes until the potatoes are tender.
In the meantime, cut shavings from half of the Parmesan cheese with a peeler. Finely grate the remaining Parmesan. Leave small basil leaves whole, cut remaining leaves into strips. Drain the chopped Pelati tomatoes in a sieve and drain well, enjoy the juice as an aperitif or use it later for a sauce or soup.
Finely blend the soup with a hand blender or in a blender jar. Return to the frying pan. Add the grated Parmesan cheese and season the soup with salt and pepper. Serve in deep cups or plates. Add a spoonful of tomatoes, a little basil and Parmesan shavings and serve the soup on the spot.