Preheat the oven to 375 degrees F.
Add the carrots, onions and garlic to a sheet tray, drizzle with the olive oil and season with a big pinch of salt and pepper. Toss everything together so it's nice and coated. Roast until tender and golden brown in spots, about 45 minutes.
Add 4 cups of the chicken broth to a saucepan and bring to a simmer. Pop the garlic out of their skins and add to the sauce pan with the roasted carrots and onions. Simmer everything together for 15 minutes. Add the basil at the very end of cooking. Puree until smooth by adding in batches to a blender or by using an immersion blender. Add the remaining broth, a little at a time, when pureeing to loosen it up a bit.
Serve in bowls with some sour cream or Greek yogurt swirled in and Grilled Cheese Dunkers on the side.
Heat a large nonstick pan over medium heat. Divide the cheese among 4 slices of the bread and top with the other 4 slices. Spread the outside of the bread with mayonnaise. Add the sandwiches to the pan and cook until golden brown and the cheese is melted, 3 to 4 minutes on each side. Slice into dunkers and serve with the soup.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.