Adjust the oven rack to the lower-middle position and preheat the oven to 325°F. Pat the chicken dry with paper towels and season with salt and pepper. Set aside.
Heat the oil in a large Dutch oven or braiser (12 – 14 inches) over medium-high heat until shimmering. Add the garlic, cut-sides down, and cook until golden brown, 2 to 4 minutes. Remove the garlic to a plate.
Reduce the heat to medium. Add the chicken and cook until lightly browned, about 4 minutes per side; transfer to a plate. Depending on the size of your pan and on the size of your chicken, you may need to sear the chicken in batches so that you don’t overcrowd the pan.
Pour off all but 1 tablespoon of the fat from the pan. Add the onion and cook over medium heat until the onion softens, about 5 minutes. Stir in flour and cook for 1 minute. Slowly stir in the broth, wine, and vinegar, scraping up any browned bits from the bottom of the pan and breaking up any lumps. Add the carrots, thyme, rosemary, and garlic. Bring to a simmer, and then remove from the heat. Nestle the chicken and any accumulated juices skin-side up in the pot. The skin should be above the surface of the liquid. Cover with a lid and transfer the pot to the oven. Braise for 1 to 1¼ hours, until the chicken is cooked through and the carrots are tender.
If the chicken isn’t crispy enough for your liking when it comes out of the oven, you can place the chicken on a rimmed baking sheet and put it under the broiler for a couple of minutes. This will help the skin crisp up nicely! Just keep a close eye on the meat because it can burn quickly under the broiler.
If you’d like to thicken the pan sauce slightly, just simmer it on the stovetop (uncovered) for 10-15 minutes, or until it reaches the desired consistency. Discard the garlic. Serve the chicken with the carrots and garnish with chopped fresh herbs.