Preheat a grill or grill pan over medium-high heat. Grill the tomatoes and half of the yellow pepper, turning occasionally, until lightly charred on all sides, about 5 minutes. Transfer to a blender along with the garlic, shallot, vinegar, salt and pepper and half of the cucumber. Puree until very smooth. Slowly add the olive oil with the motor running. Strain through a medium-meshed sieve into a bowl. Cover and chill at least 4 hours and preferably overnight. Finely chop the remaining half yellow pepper and cucumber and keep refrigerated.
Before serving, season the gazpacho with salt and pepper. Ladle into bowls and top with a little of the diced pepper and cucumber, then drizzle with olive oil and sprinkle with basil.