Gather all ingredients.

Set an oven rack about 6 inches from the heat source and preheat the oven’s broiler. Line a baking sheet with aluminum foil and place poblano peppers on the foil. Broil in the preheated oven until the skins are blackened and blistered over 80% of the surface, 8 to 10 minutes per side. Remove from the oven and turn the broiler off.

Transfer peppers to a resealable plastic bag. Seal the bag and let peppers steam and cool for about 20 minutes. Remove cooled peppers from the bag; scrape off and discard blackened skin.

Cut a slit lengthwise into the side of each pepper, then remove the seeds and veins. To reduce spiciness, rinse out any remaining seeds with water, then pat dry with paper towels.

Line a baking sheet with parchment or waxed paper. Place 1/2 cup flour into a shallow bowl for dredging.

Mix mozzarella, Monterey Jack, and Cheddar together in a bowl until thoroughly blended. Divide into four equal portions, then lightly squeeze each portion in your hands to make a cone shape. Insert each cone through the slit into a pepper, then pin the openings closed with toothpicks.

Dredge stuffed peppers in flour, then dust off any excess. Place onto the prepared baking sheet and into the freezer for at least 30 minutes while you make the sauce. Keep remaining dredging flour available for Step 10.

To make the sauce, purée stewed tomatoes in a blender and set aside. Heat 1 tablespoon of vegetable oil in a saucepan over medium heat. Add onion and sauté until translucent, about 5 minutes. Add garlic and sauté for about 30 seconds. Pour in puréed tomatoes, chicken broth, vinegar, oregano, cumin, black pepper, hot pepper sauce, and cinnamon. Reduce the heat to medium-low and simmer, stirring often, until sauce is reduced by half and thickened, about 20 minutes. While the sauce is simmering, preheat the oven to 200 degrees F (95 degrees C) and warm a platter in the oven.

To make the batter and finish the chiles rellenos, mix 1/3 cup flour with salt in a bowl. Beat egg whites in a separate bowl with an electric mixer until stiff peaks form. Lightly mix in egg yolk and flour mixture until a fluffy batter forms.

Remove peppers from the freezer and dredge in flour again. Shake off excess flour, then use a spoon to gently dip each pepper into egg batter until completely coated.

Heat frying oil in a heavy cast iron skillet over medium-high heat until shimmering. Working in batches, gently place coated peppers into hot oil and fry until batter is golden brown and cheese filling is hot, about 5 minutes per side. Transfer fried peppers to the warmed platter while you finish frying.

To serve, ladle a generous spoonful of sauce onto each plate, then place a pepper in the center.

Serve hot with a dollop of sour cream over top.