Preheat your broiler and place the poblano peppers on a baking sheet. Roast them until the skin is blackened and blistered, about 10 minutes, turning occasionally. Once cooked, transfer to a bowl and cover with plastic wrap to steam, then peel off the charred skins and remove seeds.
While the peppers are roasting, heat the olive oil in a skillet over medium heat. Add chopped onion and sauté until translucent and fragrant, about 5 minutes. Stir in minced garlic and cook for another 30 seconds until aromatic.
Once cooled, roughly chop the peeled poblano peppers and add them to a blender along with the sautéed onions and garlic. Pour in vegetable broth, add cumin, and season with salt and pepper. Blend until smooth and velvety, adjusting seasoning as needed.
Pour the poblano sauce back into the skillet and gently simmer over low heat for 5 minutes to meld the flavors and thicken slightly. Meanwhile, warm the tortillas to make them pliable.
Spread a thin layer of the smoky sauce across the bottom of your baking dish. Dip each tortilla briefly in the remaining sauce, then spoon shredded chicken and a sprinkle of cheese onto the center. Roll up the tortillas and place them seam-side down in the dish.
Pour the rest of the poblano sauce evenly over the assembled enchiladas, ensuring each is well coated. Sprinkle the remaining cheese on top for a gooey, cheesy crust.
Place the dish in the preheated oven and bake at 375°F (190°C) for about 20-25 minutes, or until the cheese is melted and bubbly with some golden brown spots.
Remove from the oven and let rest for a few minutes. Serve warm, garnished with fresh cilantro or additional cheese if desired. Enjoy the smoky, cheesy, tender enchiladas!
