For game hens, many Italian cooks would lean toward herbs with flavors that would work well with the darker meat, along with the usual garlic plus shallot or onion.
Heat 5 or 6 Tbls of evoo in a small pot over low heat, with 3 or 4 cracked bay leaves and several turns of the peppermill mixed in. Keep the heat low and heat about 10 min.
Meanwhile, mince 6 or 7 cloves of garlic (about 1.5 Tbls), 1 whole shallot (about 2-2.5 Tbls), enough fresh rosemary to equal 2 tsp, enough fresh sage to equal 1 tsp, and about 1/2 tsp fresh thyme. Mix in a small bowl with 1/2 c balsamic vin and a pinch of salt.
Allow the oil mix to cool some and mix well with the vin-herb mix.
Split your hens, put in a Ziploc, and the marinade, close the bag, turning to coat well. Marinate overnight.
Drain the hens well when ready to cook, allowing the marinade to drain into the bag. Reserve the marinade; use to baste the hens once, when they are about half cooked.