Prep the chiles. Roast and peel the Hatch Chiles -- here's How to Roast and Peel Peppers. Cut the stem end off of each one and carefully, use a small paring knife or your fingers, to make a slit in each of the peppers to open them and remove the seeds.
Preheat the oven to 375°F.
Cook the onion and garlic with corn and paprika. Coat a large sauté pan with the olive oil and place it over medium heat. Add the onion and garlic, and cook until the onions are soft and beginning to brown, about 7 minutes. Add the corn and paprika to the pan and cook for another 2 minutes or so, just to blend the flavors. Set aside.
Make the batter. In a large mixing bowl, combine the cornmeal, baking soda, sugar and salt. Make a small well in the middle and add the buttermilk and eggs. Mix until the batter is smooth. Fold in the melted butter and the onion-corn mixture.
Layer the components. Place 6 of the peppers, smooth side down, in a nonstick, 9 X 13 X 3-inch baking pan. (If your pan is not nonstick, drizzle the bottom lightly with olive oil.) Set the remaining 6 peppers aside. Pour about 2 cups of the batter evenly over the peppers and sprinkle with ¾ cup of the cheese. Gently add a layer of the remaining 6 peppers, smooth side down, on top of the cheese. Pour the remaining batter (about 2½ cups) over the peppers, and sprinkle with the remaining 1¼ cup of the cheese.
Bake. Bake in the preheated 375°F oven until it begins to brown on top and the batter is set, about 25 minutes. If the casserole is finished, but the top is not as brown as you'd like, place it under the broiler for about 30 seconds.
Cool and serve. Let the casserole cool in the pan for at least 20 minutes before serving.