In a large deep frying pan, heat the oil over moderately high heat. Season the chicken with 1/2 teaspoon of the salt and 1/4 teaspoon pepper. Put the chicken in the pan and brown well on both sides, about 8 minutes in all.
Remove chicken from pan. Pour off all but 1 tablespoon of the fat. Reduce the heat to moderately low. Add the onion and garlic and cook, stirring occasionally, until the onion starts to soften, about 5 minutes.
Add the rosemary and wine to the pan. Bring to a simmer, scraping the bottom of the pan to dislodge any brown bits. Boil until the wine is reduced to approximately 1/4 cup, 1 to 2 minutes.
Add the broth, tomatoes, olives, the remaining 1/2 teaspoon salt, and the chicken, skin-side up, with any accumulated juices. Cover and simmer until the chicken is just done, 20 to 25 minutes.
Push the chicken to the side of the pan and then mash the garlic cloves with a fork. Cracked black pepper and the basil, bring just to a simmer, and serve hot.