Preheat broiler, build a medium-hot fire in a charcoal grill, or heat a gas grill to high. If broiling, place chiles on a rimmed baking sheet. If grilling, put chiles directly on grill grate. Roast, turning occasionally, until tender and nicely charred all over, 15-20 minutes.
Transfer chiles to a large bowl; cover with plastic wrap and let steam for 15 minutes. Peel chiles. Halve lengthwise; discard seeds.
Cut crosswise into 1/4" strips.
Heat a large dry heavy skillet over medium-high heat.
Add onion; cook, stirring often, until beginning to char, 6-7 minutes.
Add garlic; cook until fragrant, about 1 minute.
Add oregano and 1 cup water; simmer until onion is tender and water has evaporated, 5-7 minutes.
Add chiles; cook until flavors meld, about 5 minutes.
Remove from heat; stir in crème fraiîhe and cheese.
Add water by tablespoonfuls if mixture is too dry. Season to taste with salt.
