Heat a medium nonstick skillet over medium heat. Add the oil and the sliced mushrooms to the hot skillet. Season mushrooms with black pepper, salt and thyme. Saute the mushrooms 10 minutes or, until dark and tender, then remove from heat. Transfer the mushrooms to a dish and return the skillet to the stove over medium heat.
Add another turn of extra-virgin olive oil to the skillet and add the onions, garlic and jalapeno pepper; saute for 2 or 3 minutes, then add beans and corn to the pan. Stir in chopped tomatoes and barbecue sauce, then season salsa with salt and pepper. Transfer warm salsa to a serving dish.
Heat a griddle pan or large nonstick skillet over medium to medium high heat. Add a drizzle of oil to the pan and 1 flour tortilla. Cook tortilla 1 minute, then turn it over. Sprinkle 1/2 cup sharp cheddar over 1/2 of the flour tortilla. Cover the cheese with 1/4 of the cooked sliced mushrooms. Fold the plain 1/2 of the tortilla over top of the filling and gently press down with a spatula. Cook the filled quesadilla 30 seconds to a minute longer on each side to lightly brown and crisp the quesadilla and melt the cheese. Remove the quesadilla to a large cutting board or transfer to a warm oven to hold, then repeat with remaining quesadilla ingredients.
Cut each quesadilla into wedges and serve with warm salsa for topping.