Roughly chop the cucumber, red bell pepper, tomatoes, garlic, and shallot. Combine in a large bowl with tomato juice, red wine vinegar, olive oil, salt, pepper, and fresh basil. Mix thoroughly, cover, and refrigerate overnight to allow the flavors to develop.

Blend the reserved cucumber trimmings and the fresh cucumber. Season with lemon juice, a pinch of sugar, salt, and pepper. Pass the liquid through a fine-mesh chinois or sieve. Bring the liquid to a boil, whisk in the agar agar, and simmer for 2 minutes. Pour into a container and refrigerate until set. Once firm, blend the jelly until it forms a smooth, pipeable gel.

Dried Tomatoes: Halve cherry tomatoes, season with salt, pepper, and garlic powder. Bake at 90°C (195°F) until shriveled and concentrated. Marinated Pepper: Finely dice peeled red bell pepper. Toss with olive oil, red wine vinegar, salt, and pepper. Garlic Chips: Thinly slice garlic and fry slowly in oil until golden and crispy. Drain on paper towels.

Transfer the marinated vegetable mixture to a high-speed blender. Process until completely smooth. For a refined texture, pass the soup through a chinois to remove any remaining pulp or skins. Keep chilled until service.

In a chilled shallow bowl, arrange a small mound of the marinated pepper dice. Place the dried tomatoes, a few fresh cherry tomato quarters, and the crispy garlic chips around the dice. Pipe small dots of the cucumber gel onto the plate. Garnish with small basil leaves. Just before serving, pour the cold gazpacho into the bowl and finish with a few drops of green herb oil.