Place the oil in a very large skillet or wide pot set over medium to medium-high heat. When hot, add the next 7 ingredients. Cook and stir until the mushrooms are very tender and any moisture has evaporated, about 7 to 8 minutes.
Spoon the mushrooms into a bowl and cool to room temperature. Now toss in the cheese, jalapeno and cilantro (or green onion).
Preheat the oven to 200°F. Set a clean and large, non-stick skillet over medium heat. Place an equal amount of the mushroom mixture on one half of each tortilla. Fold each tortilla into a half-moon shape. Cook two stuffed tortillas three minutes per side, or until the cheese is melted and the tortillas are golden brown. Set the cooked quesadillas on a baking sheet and keep warm in the oven while you cook the last two. Cut each quesadilla into 4 wedges. Serve the quesadillas with salsa and sour cream.
Note: If you would like to drizzle the quesadillas with sour cream as shown in the photo, place 1/3 cup or so sour cream in a small bowl and zap in the microwave a few seconds, until fluid.