Arrange a rack in the upper third of the oven and heat the broiler to high. (If you don’t have a broiler, see Tip.) In a large, ovenproof skillet, heat the oil over medium-high. Add the poblanos and cook, undisturbed, until browned underneath, 2 to 3 minutes. Season with salt and pepper and cook, stirring occasionally, until crisp-tender, 2 to 3 minutes. Stir in the beans, salsa verde and cilantro stems, and simmer until thickened slightly, 2 to 3 minutes. Turn off the heat, sprinkle the cheese evenly over the top, then broil until melted and browned in spots, 2 to 3 minutes. Serve right away topped with cilantro leaves and a squeeze of lime.