Rinse and dry chilies. Arrange in a single layer in a 10- by 15-inch pan. Broil 4 to 5 inches from heat, turning to char chilies on all sides, about 10 minutes total. Let stand until cool to touch, 8 to 10 minutes. Pull off and discard chili skins, stems, and seeds. Cut chilies lengthwise into thin strips.
Meanwhile, peel onions and cut lengthwise into slivers. In a 12-inch frying pan or a 5- to 6-quart pan over medium-high heat, frequently stir onions, oil, and 1 teaspoon salt until onions are lightly browned, 7 to 8 minutes. Add chilies and stir occasionally to blend flavors, about 5 minutes. Season to taste with salt.
Thinly slice cheese. See notes for assembly.