Preheat your oven to 425°F (220°C).
Heat a large oven-safe skillet over medium-high heat. Add the quartered onion, whole garlic cloves (skins on), and peeled tomatillos and cook, turning often, until charred on all sides, about 5 minutes; remove any garlic cloves earlier if they char faster.
Remove the charred onion, garlic and tomatillos from the skillet and let cool. When the garlic is cool enough to handle, peel the skins off the cloves.
Return the skillet to medium-high heat and char the poblano peppers and jalapeño, turning often, until blistered on all sides, about 5 minutes.
Remove the peppers to a bowl, cover with a plate to steam for 10 minutes, then peel off the charred skins and remove the seeds from the poblanos and jalapeño.
Add the peeled peppers, peeled garlic, charred onion and tomatillos to a blender; puree until smooth and season the salsa verde with kosher salt and pepper to taste. Set aside.
Wipe the skillet if needed and return it to high heat. Pour in the 2 1/2 cups low-sodium chicken broth and bring to a boil.
Add the halved chicken breasts and 3/4 cup white rice to the boiling broth, cover, reduce the heat to low and simmer until the chicken is cooked through and the rice is tender, about 25 minutes.
Remove the skillet from heat and shred the chicken with two forks directly in the pan.
Stir the pureed salsa verde, 4 of the torn tortillas, the corn and the chopped cilantro into the skillet, mixing gently to combine. Season with additional salt and pepper if desired.
Top the mixture with the remaining 2 torn tortillas, then evenly sprinkle the cheddar and Monterey Jack cheeses over the top.
Transfer the skillet to the preheated oven and bake 15–20 minutes, until the cheese is melted and bubbly.
Serve warm topped with sliced avocado and additional cilantro if desired.