Preheat grill to medium-high. Grill tomatoes and bell pepper, turning a few times, until they soften and the skins are blistered and charred in spots, about 8 minutes.
Transfer the pepper to a plastic bag and let it steam until cool enough to handle.
Peel off the skin; cut the pepper in half and discard the stem and seeds. Place one half in a blender. When the tomatoes are cool enough to handle, core and roughly chop.
Add the tomatoes, skins and all, to the blender. Add half the cucumber to the blender along with bread, garlic, vinegar to taste, parsley, paprika , salt and pepper.
Blend until smooth. Add oil and blend until well combined.
Refrigerate until room temperature or chilled, at least 1 hour.
Before serving, finely dice the remaining cucumber and bell pepper; stir into the gazpacho.
Garnish with basil.