Heat oven to 425 degrees with racks in the upper-middle and lower-middle positions. Line two rimmed baking sheets with aluminum foil and set a wire rack inside each to allow air to circulate around the peppers. Use a potato masher to mash half of the beans with the water in a bowl until the mixture is mostly smooth.

Heat the oil in a large nonstick skillet over medium heat until it shimmers. Add the onion and cook until softened, about 5 minutes. Add the garlic, cumin, oregano, chili powder, lime zest, salt, and cayenne; cook until fragrant, about 30 seconds.

Stir the mashed bean mixture into the skillet and cook, stirring constantly, until nearly all the liquid has evaporated, 3 to 5 minutes. Stir in the remaining whole beans and corn until warmed through, about 2 minutes. Remove from heat and fold in the Monterey Jack, cheddar, cilantro, and lime juice. Season with salt and pepper to taste.

Leaving the stems intact, cut a slit lengthwise down one side of each poblano. Place the peppers in a bowl, cover, and microwave until they are just pliable, about 2 1/2 minutes.

Gently pry open the slit in each pepper, remove the seeds, and stuff the cavities evenly with the bean and cheese mixture. Lay the peppers stuffed-side up on the wire racks.

Bake until the peppers are tender and the filling is hot, 30 to 40 minutes, switching and rotating the baking sheets halfway through. Serve hot with salsa.