Preheat the broiler. Place a rack from 6 to 8 inches from the heat source. Line a rimmed baking sheet with aluminum foil, then oil the foil or spray with nonstick cooking spray.
Husk the tomatillos, and rinse them, then cut them in half. Place them cut side down on the baking sheet. Remove the seeds from one of the jalapenos, and place all four halves on the baking sheet, cut side down. Place the onion slices on the baking sheet, and brush them with a bit of olive oil. Place the limes, cut side UP on the baking sheet.
Place the baking sheet under the broiler and broil for 8 to 10 minutes until the vegetables and lime turn golden brown. Allow to cool on the baking sheet.
While the vegetables are cooling, heat 1 tablespoon of the olive oil in a large skillet over medium high heat. Season the chicken breasts on both sides with salt and pepper. Cook the chicken for about 5 minutes on each side until the outside is golden brown and the inside is cooked through, in batches if necessary. Set the chicken on a cutting board to rest for 5 minutes.
While the chicken is cooling, transfer the cooked tomatillos, jalapenos, and onions to a food processor. Squeeze the juice from the lime halves into the food processor, toss the lime rinds, and add the garlic, ¼ cup olive oil and cilantro. Pulse to chop and combine, then process until the mixture is well incorporated and fairly smooth.
Slice the chicken breasts on the diagonal, and place each on a plate (or place them all on a large serving platter). Spoon the tomatillo sauce down the center of each sliced chicken breast.
The flavor of tomatillos ranges from mild and on the sweeter side to a bit bitter or quite tart, so it’s a bit of a crapshoot to know what the result of the dish will be. Taste as you go, and see how to compensate with other ingredients until you get the flavor you’re looking for – a little sugar helps with an overly tangy tomatillo sauce. Tomatillos might be roasted, grilled, stewed or sautéed. They can also be eaten raw.Tomatillos need to be husked before using, and the husk needs to be discarded. The tomatillos will be slightly sticky after you husk them, so give them a quick rinse under running water.