Heat the oven to 400 F / 200 C. Put the chicken, 480 ml broth and 1/2 tsp salt in a 3-quart / 2.8 L saucepan and bring to a bare simmer over medium heat. Cover, reduce to low and poach for 16 to 18 minutes, until an instant-read thermometer in the thickest part reads 165 F / 74 C.
While the chicken poaches, heat the oil in a 26 cm / 10-inch skillet over medium heat. Cook the onion for 4 minutes until soft, add the garlic and cumin for 30 seconds until fragrant, then stir in the salsa verde, green chiles and 120 ml broth. Simmer for 10 minutes until the sauce coats the back of a spoon, then take it off the heat.
Move the chicken to a bowl and shred it with two forks, then stir in 2 tbsp of the poaching liquid. Mash half the beans in a small bowl with a fork. Fold the mashed and whole beans into the chicken with 100 g of the cheese, 15 g of the cilantro and 1/3 cup (80 ml) of the sauce.
Spread 1 cup (240 ml) of sauce over the bottom of a 23 x 33 cm / 9 x 13-inch baking dish. Wrap 6 tortillas in a damp paper towel and microwave for 45 seconds, until they bend like cloth. Mound about 1/3 cup (65 g) of filling down the center of each, roll tightly and set seam-side down in the dish. Warm the next 6 tortillas and repeat, packing the rolls snugly. Don't warm all 12 at once, because cold corn tortillas crack.
Pour the remaining sauce over the rolls so the tortilla ends are covered, then sprinkle with the remaining 40 g cheese. Bake uncovered for 20 minutes, until the sauce bubbles at the edges and the cheese is spotted brown. Rest for 5 minutes.
Stir the yogurt, lime juice and 1/4 tsp salt together. Spoon the crema over the enchiladas, scatter with the remaining 15 g cilantro and serve 3 per plate.