Heat a grill pan over high heat. Place whole chilies on grill and char all over, about 10 minutes. Remove from heat and split chilies. 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 one 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, then transfer to a cutting board. Repeat.
Pile up two completed quesadillas at a time and cut into three generous wedges. The yield will be 12 pieces from four 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.