Peel and cut each garlic clove in half. Remove the green shoot inside, if there is one. Fill a small saucepan with cold water and add the garlic. Bring the water to a boil. When the water begins to boil lift the garlic out of the water and into a bowl of ice water to quickly cool it. Repeat this twice, always starting with cold water in the saucepan.
Peel the onion and cut in half, then cut into large chunks. Peel the cucumber. Cut in half, then into large pieces. Halve the bell peppers, then remove the seeds and white membranes. Chop the bell peppers, then aside along with the cucumber and onion. Cut the tomatoes into large wedges and put in a bowl with the onions, cucumbers, and bell peppers.
Add the bread, torn into pieces, then pour over the water. Process everything together using a hand-held blender, or using a food processor. Strain the gazpacho through a fine-mesh strainer.
Add the oil, vinegar, and mayonnaise, then whisk or blend the soup until smooth and creamy. Season with salt and pepper.
Chill in the fridge before serving (at least 2 hours). Serve the gazpacho in soup bowls with croutons, plus an extra drizzle of olive oil.