Preheat the broiler. Place whole poblano chiles on a baking sheet and broil, turning occasionally, until charred all over, about 5 minutes per side.
Transfer charred chiles to a paper bag or sealed plastic bag and let steam for 20 minutes to loosen the skins.
Remove chiles from the bag, peel off the skins, remove stems and seeds, and chop the flesh into roughly 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.
Preheat the oven to 375°F (190°C). Grease a large baking dish or a 12-inch cast-iron skillet and add the drained pasta.
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.
Gradually 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 the mustard powder, cayenne, ground cumin, lime zest, chopped cilantro, and chopped poblano chiles. Season with Kosher salt and black pepper to taste.
Stir in half of the grated white cheddar until melted and smooth; if the sauce becomes too thick, thin with a teaspoon of milk at a time; if cheese won't melt, return to low heat briefly.
Pour the cheese sauce over the cooked pasta in the prepared dish, sprinkle with the remaining cheddar, and bake uncovered at 375°F for about 20 minutes, until browned and bubbling.
Remove from oven, sprinkle Cotija cheese over the top, and serve immediately.