Preheat the oven broiler. Place whole poblano chiles on a baking sheet and broil, turning as needed, until the skins are blackened, about 5 minutes per side.
Transfer the hot chiles to a paper bag or sealed plastic food-storage bag and let steam for 20 minutes.
Remove chiles from the bag, peel off and discard the skins, then remove stems and seeds. Chop the peeled chiles into about 1-inch pieces.
Bring a large pot of salted water to a boil and cook the elbow pasta until al dente according to package directions. Drain and set aside in the prepared baking dish or cast-iron skillet.
Preheat the oven to 375°F (190°C). Grease a large baking dish or a 12-inch cast-iron skillet and place the drained pasta in it.
In a saucepan over low heat, melt the butter. Add the minced garlic and cook about 1 minute until fragrant.
Whisk in the flour and cook, stirring, until it forms a pale, toasty paste, about 1 minute.
Slowly whisk in the milk and cook, stirring, until the sauce thickens but remains pourable, about 1–2 minutes.
Remove the sauce from heat and stir in mustard powder, cayenne, cumin, lime zest, chopped cilantro, and the chopped poblano chiles. Season with Kosher salt and black pepper to taste.
Whisk in half of the grated white cheddar until melted and fully incorporated; if the sauce is too cool to melt the cheese, return to low heat briefly, stirring constantly. If the sauce becomes too thick, thin with milk a teaspoon at a time.
Pour the cheese sauce over the pasta in the baking dish, sprinkle the remaining cheddar on top, and bake uncovered at 375°F for 20 minutes or until golden and bubbling.
Remove from oven, sprinkle with Cotija cheese, and serve immediately.
