Preheat the oven to 400°F. Line a large, rimmed oven and broiler-safe baking sheet with foil and spray it with oil.
Rinse and dry the poblanos. Cut a thin slice off the tops and remove the core and seeds. Cut a slit all the way down the side of each pepper. Set aside.
Prepare the filling: Heat the olive oil in a large, deep skillet over medium-high heat. Add the tomatoes, onion, garlic, salt, oregano, and cumin. Cook, stirring often, until the liquids have evaporated, 5-7 minutes. Off heat, stir in the chicken, mozzarella, and cilantro, mixing well.
Divide the filling among the peppers, adding it from the top and pressing it to fill the entire pepper. Pack the filling into each pepper to prevent it from spilling onto the baking sheet.
Place the peppers on the prepared baking sheet, slit side up. Lightly spray them with olive oil. Bake until the poblanos are soft and charred in places, about 30 minutes (the smell will be amazing!).
Remove the baking sheet from the oven and switch the oven to broil. Top the peppers with the cheddar cheese, sprinkling the cheese on the open part where you've cut the slit.
Broil the peppers 6 inches below the heating element (not directly below) just until the cheese is melted, 1-2 minutes. Let them rest for 5 minutes before serving.
Video
Notes
Protecting your hands with gloves while handling poblano peppers is a good idea.
In addition to oregano and cumin, I sometimes add a teaspoon of smoked paprika to the filling.
If you can only find small poblanos, you can use them to make this recipe, using 8-10 peppers instead of four. You can also mix different pepper sizes and bake them all together.
If you don't like spicy food, you can make this recipe with bell peppers.
You can keep the leftovers in a sealed container in the fridge for 3-4 days. Reheat them in the microwave, covered, at 50% power, or uncovered in a 350°F oven until heated through.
To freeze the leftovers, place them on a wax-paper-lined platter, not touching, until they are frozen, and then transfer them to freezer bags (in a single layer). You can keep them in the freezer for up to three months. Thaw them overnight in the fridge before reheating them in the oven or microwave.