Preheat your oven to 400°F (200°C).
On a parchment-lined baking sheet, toss the cherry tomatoes and whole garlic cloves with olive oil, sea salt, and black pepper.
Roast the vegetables for 20 minutes until the tomatoes have burst and are slightly charred on the edges.
While the tomatoes roast, poach or pan-sear the chicken breast until it reaches an internal temperature of 165°F, then shred it finely with two forks.
Transfer the roasted tomatoes, garlic, and any juices from the pan into a high-speed blender.
Add the vegetable broth and fresh basil to the blender, then process until the mixture is completely smooth and velvety.
Pour the soup into a small pot over low heat and stir in the Greek yogurt until fully incorporated and creamy, being careful not to let it boil.
Ladle the soup into a bowl, pile the shredded chicken in the center, and serve with a toasted slice of sourdough bread.