Peel, wash and halve the potatoes and cook in boiling salted water for about 20 minutes. Heat 1 tablespoon of oil. Roast the pine nuts in it while turning. Cut tomatoes into thin strips. Wash basil, shake dry, pluck leaves from the stalks.
Finely crush 2 tablespoons of oil and basil with a blender. Use a peeler to cut off a few slices of the Parmesan cheese for garnishing. Grate the remaining cheese into thin strips. Bring milk, tomatoes and pine nuts to the boil.
Drain the potatoes, add the milk mixture and basil puree and mash into puree. Fold in parmesan and season with salt and pepper. Serve and garnish with parmesan slices.