*Fresh green chilies, often called pasillas; available at Latin American markets and some supermarkets.
Heat 3 tablespoons oil in heavy large skillet over medium heat. Add chilies. Cover and cook until chilies are soft, adding water by tablespoonfuls if chilies become too dark, stirring occasionally, about 10 minutes. Uncover. Add garlic; sauté 1 minute. Transfer to medium bowl. Stir in cilantro. Season chili mixture to taste with salt and pepper.
Heat 2 teaspoons oil in each of 2 heavy large skillets over medium-high heat. Add 1 tortilla to each skillet. Top each with 1/4 of cheese and 1/4 of chili mixture, then 2 tablespoons salsa. Top each with 1 tortilla. Cook until bottoms are golden brown, about 3 minutes. Turn quesadillas over. Cook until bottoms are golden brown and cheese melts, about 3 minutes. Transfer to cutting board. Cut each quesadilla into 6 wedges. Transfer to platter. Repeat with remaining oil, tortillas, cheese, chili mixture, and salsa. Serve hot.