Add the olive oil to a large Dutch oven over medium heat. Season the chicken with salt and pepper. When the oil is hot, place the chicken pieces skin-side down in a single layer in the pot. Cook on both sides, turning several times, until the skin is browned, about 4 minutes. Remove to a plate and set aside.
Add the garlic, carrot chunks and onion quarters, followed by the wine, stock, apricots and rosemary. Stir together, then return the chicken to the pot. Bring to the boil, cover and braise in the oven until the chicken and vegetables are tender, about 2 1/2 hours.
Serve the chicken over Cheese Grits with the onions, carrots, garlic and apricots, making sure to get a spoonful or two of the cooking liquid. Garnish with parsley and serve.
Bring the chicken broth to a boil in a saucepan over medium heat. Slowly stir in the grits, then return to a boil and simmer for 3 to 4 minutes.
Turn down the heat to low and stir in the Cheddar and cream cheese. Cook until the Cheddar is melted and the cream cheese is incorporated into the mixture, another 3 to 4 minutes. Stir in the butter and keep warm until ready to serve.