Heat the broiler to high.
Char the poblanos under the broiler to blacken their skins on all sides, 7 to 8 minutes. Place in a bowl, cover with plastic wrap, and let stand 10 minutes.
Peel and seed the peppers, then slice. Heat a skillet over medium-high heat. Add a drizzle of extra-virgin olive oil and add a tortilla, cover the tortilla with 1/2 cup cheese and 1/4 of the peppers. 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.