Put the breadcrumbs in a small bowl and stir in 1/3 cup cold water to moisten them. Set aside to let the breadcrumbs absorb the water, about 10 minutes.
Put the almonds and garlic in a food processor and process until finely chopped. Add the moistened breadcrumbs. Pulse for a moment to mix and then add the green pepper, the sliced cucumber, the 1/4 cup basil, and the vinegar. Process for several minutes until the mixture is as finely puréed as possible. With the machine running, gradually add the olive oil.
Transfer the mixture to a large bowl and stir in 1-1/4 to 1-1/2 cups cold water (use ice water if serving immediately). Season with salt and pepper and chill for several hours or overnight. Before serving, taste and add more salt and pepper, if necessary.
For each serving, put an ice cube in a chilled soup bowl and ladle in the gazpacho. Garnish with the diced cucumber, grape halves, and torn basil leaves.