Char chiles on a gas burner or broil in the oven, turning often, until blackened all over, about 10 minutes. Put in a bowl, cover, and let steam 20 minutes.
Using a paper towel, peel blistered skin off chiles. Stem and seed chiles, then slice into 1/2-in. strips lengthwise. Halve any pieces longer than about 3 in. Season with salt.
Pull Oaxaca cheese apart into pieces about 1/2 in. wide and 3 in. long. If using Monterey jack, cut into 1/2- by 3-in. sticks.
Make ahead: Up to 2 days, char and peel chiles and prep cheese; keep chilled.