Peel and chop cucumber. Chop tomatoes. Chop bell pepper, removing the pulpy ribs and seeds. Slice the whites of the green onions. Tear bread into small pieces. Reserve some cucumber and other vegetables for garnish, if you like.
Place all ingredients except crème fraîche and tomato juice in a large bowl. Season with salt and pepper. Mix by hand, getting the bread well soaked with the juices. Allow to stand for half an hour in the refrigerator.
Add soaked ingredients to food processor. Pulse and then blend until smooth. This may have to be done in batches, depending on the size of the food processor bowl.
When all ingredients are well blended, force the purée through a the sieve until only skins, pulp and seeds remain. Dispose of the remnants, which should be quite dry and juiceless. The strained soup should have a creamy consistency when stirred together.
Taste for seasoning and consistency. Adjust with salt and pepper and add tomato juice if the soup is too thick for your liking. Add a drizzle of olive oil, if needed.
Chill before serving to cellar temperature (cooler than room temperature, but not ice cold). Serve with optional garnishes and toast.