Place chicken breasts on a cutting board and trim off excess fat. Cover the chicken with a large sheet of saran wrap or wax paper. Using a meat mallet or the bottom of a heavy skillet, pound the chicken until it is a uniform 1/2 inch thick. (This helps tenderize the chicken and makes it cook more evenly).
Transfer the chicken to a large bowl or ziplock bag. Add the remaining marinade ingredients and toss until well coated. Cover and refrigerate for at least 1 hour, up to 24 hours, until ready to grill.
Place the tomatillos, cut onion, and jalapeno on a small skillet and drizzle with oil.
Heat a grill over medium-high heat. On one half of the grill, place the skillet with the tomatillos. Place the chicken on the other half of grill. Grill, with the grill closed, until the chicken is cooked through-- about 4 minutes a side. Grill the veggies on the skillet until they begin to blister and are just tender, about 15 minutes total, rotating every 3 to 4 minutes. Remove from heat and set aside.
While the chicken rests before slicing, make the salsa. Place that charred tomatillos, onions, and jalapeno (remove the stem) into a food processor or blender with the cilantro, salt, cumin, and lime. Blend until just smooth, transfer to a bowl for serving.
Slice the chicken and serve in warmed tortillas with sliced avocado, diced onion, cilantro, and top with the tomatillo salsa! Serve with a wedge of lime and enjoy!