Peel, wash and cut the potatoes into pieces. Peel shallots and garlic, dice finely. Heat 1 tablespoon of oil in a large pot. Sauté shallots and garlic in it. Steam the potatoes briefly.
Add wine and stock. Cover and simmer for about 20 minutes.
Grate the parmesan, add to the soup and melt in it while stirring. Wash the tomatoes and dab dry. Heat 1 tablespoon of oil in a frying pan. Fry the tomatoes for 3-4 minutes while turning them, dust with icing sugar and caramelise slightly.
Whip cream until semi-stiff and stir into the soup. Puree the soup with a hand blender. Season to taste with salt and pepper. Serve the soup with the tomatoes. Stir in 1 tsp. pesto each.