Adjust an oven rack 6 inches from the broiler element and heat the broiler. Place the poblanos on an aluminum foil–lined rimmed baking sheet and broil, turning as needed, until the skins are charred and blistered, 15 to 20 minutes. Transfer the peppers to a large bowl, cover with plastic wrap, and let them steam for 5 minutes. Once cooled, remove the skins, stems, and seeds, then chop the peppers into half-inch pieces. Reduce the oven temperature to 400°F and move the rack to the middle position.

Combine the tomatillos, half of the onion, half of the cilantro, broth, cream, a portion of the oil, whole garlic cloves, lime juice, sugar, and salt in a food processor. Process until the sauce is completely smooth, about 2 minutes, then season with salt and pepper to taste.

Mash half of the beans in a large bowl with a potato masher or fork until mostly smooth. Heat oil in a large skillet over medium heat until it shimmers, then sauté the remaining onion until softened and lightly browned, 5 to 7 minutes. Stir in the minced garlic, chili powder, coriander, and cumin and cook until fragrant. Fold in the chopped poblanos, mashed beans, and remaining whole beans and cook until the mixture is warmed through.

Transfer the bean mixture to a bowl and stir in half of the Monterey Jack, a splash of the pureed tomatillo sauce, and the remaining cilantro. Spread a thin layer of the sauce over the bottom of a 13 by 9-inch baking dish. Brush both sides of the tortillas with oil, wrap them in a damp dish towel, and microwave until they are warm and pliable, about 1 minute.

Spread a portion of the bean-cheese filling across the center of each warm tortilla, roll tightly, and place seam-side down in the baking dish. Pour the remaining sauce over the top to cover completely and sprinkle with the remaining cheese. Cover the dish tightly with greased foil and bake until the enchiladas are heated through, about 25 minutes. Let the dish rest for 5 minutes before serving so the sauce can set.