Preheat oven to 350 degrees F. Slice the root end off the entire head of garlic and turn it upside down, wrap it in aluminum foil and mold the foil so it supports the garlic to allow the cut end to stand upright in the oven. Pour 1 tablespoon extra-virgin olive oil on the garlic so it slips between the skin and the cloves and wrap with the foil so it is enclosed. Roast for about 1 hour and set aside until it is cool enough to handle.
To make the olive salsa for the chicken topping, in a small bowl combine the tomato, basil, parsley, peppercorns, olives and 3 tablespoons extra-virgin olive oil. Gently squeeze the roasted garlic cloves from the skins into the mixture. Fold everything together and set aside to let flavors integrate.
For the rice accompaniment to the chicken, melt the butter in a medium saucepan over medium heat and stir in the basmati rice, thyme leaves, salt, and pepper, allowing rice to toast for a few minutes. Then add the chicken stock, bring to a boil, reduce heat to low, cover and let simmer covered and undisturbed for 20 minutes. Remove from heat and let stand 5 minutes.
In a large saute pan, heat the chicken broth over medium high heat. Add the basil, tarragon and parsley, bring to a boil to integrate flavors, then reduce heat. This is your poaching liquid. Season the chicken breasts generously on both sides with salt and pepper, rubbing the seasoning in, and place the chicken in simmering poaching liquid. (Remember, you are poaching, not boiling. The liquid should be kept at a simmer.) Poach chicken until cooked through, about 10 minutes, turning breasts midway through cooking period. Remove to paper towels to drain.
Spoon some basmati rice onto serving plate, top with a piece of chicken, and spoon olive salsa over chicken. Garnish with baby carrots.