Preheat the oven to 375°F (190°C) and position the oven rack for broiling the poblanos first.
Roast the whole poblano chiles under the broiler until evenly blackened, about 5 minutes per side.
Place the hot chiles in a paper bag or sealed food-storage bag and let steam for 20 minutes to loosen the skins.
Remove the chiles from the bag, rub off the charred skin, then remove stems and seeds and chop into ~1-inch pieces.
Bring a large pot of salted water to a boil, cook the elbow pasta until al dente according to package directions, then drain and set aside.
Grease a large baking dish or a 12-inch cast-iron skillet and add the drained pasta to it.
In a saucepan over low heat, melt the butter, add the minced garlic, and cook 1 minute until fragrant.
Whisk in the flour and cook about 1 minute until it forms a light, toasty paste.
Slowly whisk in the milk and cook, stirring, until the sauce thickens but is still pourable, about 1–2 minutes.
Remove the sauce from the heat and stir in mustard powder, cayenne, cumin, lime zest, chopped cilantro, and the chopped poblano chiles; season with salt and black pepper to taste.
Stir in half of the grated white cheddar until melted and smooth, returning to low heat briefly if needed to help the cheese melt; if the sauce becomes too thick, thin with a little milk, 1 teaspoon at a time.
Pour the cheese sauce over the pasta, top with the remaining cheddar, and bake uncovered in the preheated oven for 20 minutes or until bubbly and golden.
Remove from the oven, sprinkle with Cotija cheese, and serve immediately.