Preheat broiler to high. Place poblanos on a baking sheet and char under the broiler, turning, until the skin blackens on all sides, 7 to 8 minutes.
Place the charred peppers in a bowl and cover with a towel or plastic wrap; let stand 10 minutes. Peel and seed the peppers, then slice.
Heat 1 tablespoon oil in a large skillet over medium heat. Add the onion and cook until tender, 4 to 5 minutes. Remove from skillet and mix with the peppers.
In the same skillet over medium-high heat add a drizzle of the remaining 3 tablespoons oil and one tortilla, cover the tortilla with one-quarter of the cheese, poblano and onion mixture and cilantro.
Fold the tortilla in half and cook 2 minutes to crisp and char the tortilla evenly and melt cheese. Remove and cut into wedges; repeat with remaining oil, tortillas and filling ingredients.