Step 1Heat the oil and garlic in a saucepan over very low heat just until tiny bubbles appear around the garlic, about 5 minutes. Remove the pan from the heat and let the oil stand 1 hour. Use a slotted spoon to discard the garlic.
Step 1Build a charcoal fire in an outdoor grill and let it burn until the coals are covered with white ash. In a gas grill, preheat on high.
Step 2Lightly oil the cooking rack. Brush the mushroom caps liberally with garlic oil and lightly season them with salt and pepper. Place the mushrooms on the grill and cover. Grill, turning once, until the mushrooms are tender, about 6 minutes. Transfer them to a cutting board and slice into 1/2-inch-wide strips.
Step 3Place 4 tortillas on a work surface and sprinkle each with 1/4 cup of the cheese. Top with equal amounts of the mushroom strips, sprinkle with the herbs, then carefully place each on the grill. Top each with one of the remaining tortillas. Grill the quesadillas until the underside is lightly browned, about 45 seconds, then turn and grill to brown the other side. Cut each quesadilla into 6 wedges and serve immediately.