Cut poblano in half lengthwise; discard seeds and membranes. Place poblano halves, skin sides up, on a foil-lined baking sheet; flatten with hand. Broil for 8 minutes or until blackened. Wrap tightly in foil. Let stand 5 minutes. Peel; coarsely chop. Place poblano in a bowl.
Place chicken in a skillet, and cover with water; bring to a boil over medium-high heat. Boil 1 minute. Remove pan from heat; cover and let stand 15 minutes or until chicken is done. Remove chicken from pan; cool slightly. Shred chicken with 2 forks. Add chicken to poblano.
Heat a large nonstick skillet over medium-high heat. Add 1 teaspoon oil; swirl to coat. Add mushrooms; sauté 5 minutes or until tender. Add onion and garlic; cook 2 minutes. Add mushroom mixture to poblano mixture. Stir in oregano, salt, and red pepper. Top one side of each tortilla with 2 tablespoons cheese and about 1/3 cup poblano mixture. Fold tortillas in half over filling.
Wipe pan clean with paper towels. Heat pan over medium heat. Add 1 teaspoon oil; swirl to coat. Add 4 quesadillas to pan, and cook 1 minute on each side or until golden. Remove quesadillas from pan, and cut each into 3 wedges. Repeat procedure with remaining 1 teaspoon oil and remaining 4 quesadillas.
Combine rind and yogurt in a small bowl, stirring with a whisk. Serve yogurt mixture with quesadillas.