Preheat Oven: Preheat your oven to 400°F (200°C). Line a large, rimmed, oven- and broiler-safe baking sheet with foil and spray it with olive oil.
Prepare the Peppers: Rinse and dry the poblano peppers. Cut a thin slice off the tops and remove the core and seeds. Cut a slit down the side of each pepper to open them up. Set aside.
Make the Filling: In a large, deep skillet, heat the olive oil over medium-high heat. Add the diced tomatoes, onion, garlic, salt, oregano, and cumin. Cook, stirring frequently, until the liquids have evaporated, about 5-7 minutes. Remove the skillet from the heat and stir in the shredded chicken, mozzarella, and chopped cilantro. Mix well to combine.
Stuff the Peppers: Divide the filling evenly among the peppers, adding it from the top and pressing it to fill the entire pepper. Make sure the filling is packed into each pepper to prevent it from spilling out during baking.
Bake the Peppers: Place the stuffed peppers on the prepared baking sheet, slit side up. Lightly spray them with olive oil. Bake in the preheated oven until the poblanos are soft and slightly charred in places, about 30 minutes. The aroma will be incredible!
Add the Cheese: Remove the baking sheet from the oven and switch the oven to broil. Top each pepper with the shredded cheddar cheese, sprinkling it over the open slit.
Broil and Serve: Broil the peppers 6 inches below the heating element (not directly underneath) just until the cheese is melted, about 1-2 minutes. Let the peppers rest for 5 minutes before serving.