Place whole chiles on grill and char all over. about 10 minutes.
Remove from heat and split chiles. Scrape away seeds with a spoon and slice.
Heat a large nonstick skillet or griddle over medium high heat. Char the tortilla and blister it on 1 side, 20 seconds, then flip tortilla. Cover half of the tortilla with cheese and chilies, then fold over. Press down gently with spatula. Cook quesadilla 15 seconds more on each side, transfer to a cutting board. Repeat. Pile up 2 completed quesadillas at a time and cut into 3 generous wedges. The yield will be 12 pieces from 4 quesadillas.
Serve slices on a large platter with small dishes of salsa verde and sour cream for topping.
Garnish the platter and toppings with chopped cilantro.
