In a food processor fitted with a sharp metal blade, purée the cilantro, garlic, artichoke hearts, salt and 1 cup of the grated Parmesan cheese until smooth.
With the motor running, add the garlic oil in a slow, steady stream.
Pour this mixture into a medium bowl; prick the chicken all over with a fork and then marinate the chicken in the mixture for at least 2 hours.
In a heavy saucepan, mix together the water, milk and polenta; cook over low heat, stirring constantly, for about 30 minutes or until it is thick, and then stir in the butter and ½ cup of the Parmesan and season to taste with salt and pepper; keep warm.
Grill the chicken over medium coals, basting with the remaining marinade, until cooked; cut into bite-size pieces and keep warm.
Divide the polenta evenly among 4 large warmed dishes.
Place the chicken on top and sprinkle with balsamic vinegar and basil-flavoured oil.
Sprinkle more grated Parmesan and snipped chives on top.