In a medium skillet, HEAT 1 Tablespoon oil over low heat. ADD onion. STIR and COOK until soft, approximately 10-12 minutes.
ADD sugar and balsamic vinegar. COOK for another 5 minutes or until the onion is caramelized. REMOVE onions from the pan onto a plate/bowl and SET ASIDE.
HEAT butter and remaining oil in the skillet over medium heat. ADD garlic, mushroom and thyme. STIR and COOK for 3-4 minutes until mushrooms are soft. REMOVE pan from the heat and SET ASIDE.
HEAT a grill or grill pan over high heat. TOAST the tortillas, one at a time, until slightly charred.
PLACE 4 tortillas, toasted-side up, on a cutting board.
TOP each tortilla with ¼ each of the caramelized onion, mushroom mixture, grated cheese arugula/spinach. TOP with the remaining tortillas, toasted side down, to created quesadilla.
HEAT the grill pan back to high heat. ADD each quesadilla to the heated grill pan. COOK for 2-3 minutes or until toasted on one side.
TURN quesadilla over and COOK for another 1-2 minutes until the tortilla is toasted and cheese is melted.
To make the lemon sour cream:
COMBINE the sour cream and lemon juice in a small bowl. MIX to combine thoroughly. SERVE with quesadillas.