To a large baking sheet (line with parchment paper if desired) add the cherry tomatoes, large tomatoes, quartered yellow onion, halved shallots and halved head of garlic.
Drizzle the olive oil, salt, pepper, chili flakes, onion powder and garlic powder over everything on the sheet pan and gently toss to coat everything on the pan. Roast for 30-40 minutes, until everything is tender and starting to turn golden. Remove from the oven and set aside.
While the tomatoes are roasting, prepare the chicken. Season the chicken breasts well with salt and pepper on both sides. Add 1 tbsp olive oil to an oven-safe frying pan and heat oil over medium-high heat for 2 minutes. Add the seasoned chicken breasts and turn the heat down to medium-low, cook the chicken breasts for 3 minutes, flip and cook for another 3 minutes. Turn the heat off and place the chicken in the oven (while the tomatoes are cooking) for 10-12 minutes or until the chicken is cooked through to 165F. Let the chicken breasts rest on a plate until cool enough to handle.
Once the tomatoes have cooled slightly and the garlic is cool enough to touch, begin removing each garlic clove from the bulb and place on the sheet pan with the tomatoes.
Into a blender add the 1/2 cup half and half cream, 1/2 cup Boursin cheese, 1/2 cup chicken broth, the garlic cloves (removed from the bulb) along with everything else on the sheet pan, both of the chicken breasts (chopped) and 1/2 tsp kosher salt. Blend on high for 30 seconds to 1 minute in a high speed blender until everything is combined.
Serve the soup in bowls with a drizzle of olive oil, spoonful of boursin cheese and minced chives.