Broil the poblanos on a sheet pan, turning for 10 to 12 minutes until blistered. Cover in a bowl for 10 minutes, then peel, seed and slice into strips; pat dry.
Heat the oven to 350°F. Oil a 9-by-13-inch baking dish and layer half the tortilla strips, beans, corn, peppers and cheese; repeat with the remaining halves.
Whisk the eggs, milk, cumin and salt and pour evenly over the layers. Press down gently so the tortillas contact the custard.
Cover with foil and bake for 30 minutes. Uncover, add the crushed chips and bake for 15 to 20 minutes more, until the center reaches 160°F.
Rest for 10 minutes so the layered tortilla filling holds when cut.