Cut the chiles in half the long way, seed them, and arrange them on a foil-lined baking pan skin side up. Brush the skin with olive oil. Set under a 500F broiler and broil until the skin is mostly black and bubbling away from the flesh. Set aside.
Heat the butter in a large skillet over medium-high heat. When the butter is sizzling, add the chicken thighs. Brown each side, about two minutes each. Add 1 cup chicken broth, and bring to a boil. Cover the skillet with foil, and turn the heat to medium. Poach the thighs until cooked through, about 5 minutes. Set aside.
The chiles should have cooled by now. Using your fingers or a couple of forks, separate the skins from the flesh of the chiles. (They should peel off fairly easily). Discard the skins and roughly chop the chiles.
Add 2 tbsp olive oil to a pot or dutch oven over medium heat. When olive oil is shimmering, add the onions, garlic, cumin, oregano, salt and pepper. Cook until the vegetables, about 6 minutes. Add the chopped chiles, the diced tomatoes, and 1 1/2 cup of chicken broth. Bring to a boil, then reduce to a simmer and cook for 20 minutes.
Mash the green chile mixture until it's a consistency you'd like. Add the chicken and let the chicken heat up to the temperature of the sauce, about 2 minutes.
Preheat an oven to 350 degrees.
In an 8-inch pie pan, spread a thin layer of the Hatch chile sauce. Add a layer of corn tortillas. Add another layer of Hatch chile sauce, then a layer of cheese. Add another layer of corn tortillas, layer of Hatch chile sauce and on until you're final layer of tortillas, chile sauce and cheese is at the top of the pan. (Cheese should be the very top layer). Pour in about 1/2 cup chicken broth and put in the oven for about 30 minutes.