1. Heat Broiler to High (500º). Remove and discard the papery husk from tomatillos and slice the tomatillos in half. Toss tomatillos on a baking sheet with 1/2 tbsp olive oil. Season with salt & pepper. Char the tomatillos under broiler for 4-5 minutes, until charred and soft. Turn off broiler and reduce oven to 400º.
2. Zest and halve the lime.
3. Mince garlic.
4. Mince jalapeño, removing ribs and seeds. Mince shallot.
5. Combine 1/2 tbsp olive oil, half of minced garlic, half the lime zest and pinch of salt & pepper.
6. Rub the chicken breasts with oil mixture.
7. Heat large pan over medium-high heat. Sear the chicken for 2-3 minutes per side, until golden brown.
8. Transfer chicken to baking sheet and place in oven to finish for 10-12 minutes. Let rest 5 minutes, then slice thinly.
9. Chop tomatillos and cilantro.
10. Halve, pit and cube avocado.
11. In a medium bowl, toss the avocado, tomatillos, lime zest, garlic, shallot, 1/2 of jalapeño, cilantro, and lime juice. Season with salt & pepper.
12. Serve chicken topped with avocado salsa verde.