Preheat the broiler. Place the poblanos on a baking sheet and broil, turning, until blackened. Transfer to a bowl; cover. Let stand until cool enough to handle. Peel off the skins and remove the seeds; discard. Chop into 1/2-inch pieces.
Using a food processor, mix the tomatillos, jalapeno, onion, cilantro, lime peel, lime juice and honey until smooth; season with salt.
In a large bowl, season the beef with the cumin, salt and pepper; combine. Mix in the poblanos. Halve the meat mixture and form each into four 3-inch patties.
In a cast-iron skillet, heat the oil over high heat. Cook the patties, turning once, about 5 minutes for medium-rate. During the last minute of cooking, top with the cheese and tent with foil. Dot the roll tops with the tomatillo sauce and assemble the sliders. Serve with the extra sauce.