Heat oil in a large frypan over medium-high heat. Cook basil in oil until it begins to get crispy and has a strong basil smell; this will take several minutes. Scoop basil out of oil with a slotted spoon and drain on paper towels.
Remove half of the remaining oil from the pan. Sprinkle chicken with salt and pepper to taste and brown in oil; this should take about 4 minutes per side.
Stir in chicken stock, wine and balsamic vinegar. Season with bay leaves and thyme. Cover and simmer over low heat until chicken is done and sauce is reduced; this should take about 25 minutes.
Meanwhile, in a small saucepan combine water and rice. Bring to a boil. Reduce heat to low, cover and cook for 20 minutes.
Place halved tomatoes in the pan with the chicken, cover and simmer until the tomatoes are warm. Spoon chicken pieces, tomatoes and sauce over rice and garnish with fried basil.