Arrange the peppers skin-side uppermost in a foil-lined grill pan, brush with 2 tablespoons of the oil, then grill for about 10 minutes until the skins are blackened and the peppers softened. Wrap the foil around the peppers, then leave them to cool for 10 minutes.
Meanwhile, heat the remaining oil in a saucepan, add the onion and fry gently for 5 minutes until softened and just beginning to brown. Mix in the garlic and cook for 1 minute, then mix in the tomatoes, stock, vinegar and salt and pepper.
Peel the blackened skins off the peppers then roughly chop the peppers. Add to the saucepan then bring to the boil, cover and simmer for 30 minutes. Allow to cool slightly, then purée in batches in a blender or food processor until smooth.
Return the soup to the pan, reheat then taste and adjust the seasoning if needed. Cut the stifato sticks or bread rolls into long strips, then lightly toast on both sides. Spread with the pesto then sprinkle with the cheese and grill until just melting. Ladle the soup into bowls and drizzle with some olive oil, a few basil leaves and some black pepper.