Adjust oven racks to upper-middle and lower-middle positions and heat oven to 425°F. Line two rimmed baking sheets with aluminum foil and set a wire rack in each to help heat circulate around the peppers.

Mash half of the beans and the water together in a bowl using a potato masher until the mixture is mostly smooth.

Heat oil in a 12-inch nonstick skillet over medium heat until shimmering. Add the onion and cook until softened, about 5 minutes.

Stir in the garlic, cumin, oregano, chili powder, lime zest, salt, and cayenne. Cook until fragrant, about 30 seconds.

Stir in the mashed bean mixture and cook, stirring constantly, until nearly all liquid has evaporated, 3 to 5 minutes.

Stir in the remaining whole beans and the corn. Cook until warmed through, about 2 minutes.

Remove the skillet from the heat. Stir in the Monterey Jack, cheddar, cilantro, and lime juice. Season with salt and pepper to taste.

Cut a slit lengthwise down one side of each poblano, leaving the stem intact. Place the peppers in a covered bowl and microwave until just pliable, about 2 1/2 minutes.

Gently pry open the poblanos and remove the seeds. Stuff the peppers evenly with the bean and cheese mixture.

Lay the poblanos stuffed-side up on the prepared wire racks. Bake until the peppers are tender and the filling is bubbly, switching and rotating the sheets halfway through, 30 to 40 minutes.

Serve hot with salsa.