Preheat the grill to medium. Heat the oil and butter in a large saute pan over medium heat. Add the onions and slowly cook until caramelized, about 45 minutes. Then stir in the balsamic vinegar and thyme. Let cool slightly before using in quesadilla.
Place 6 quesadillas on a flat surface and divide the cheese and onions among the tortillas. Season filling with salt and pepper. Stack the tortillas to make 3 (2-layer) tortillas, and top with the remaining tortillas. Brush the tops with oil and grill, oiled-side down, until golden brown, about 2 to 3 minutes. Flip the quesadillas, close the cover, and grill until the cheese has melted, about 5 minutes. Slice into quarters, and place 3 quarters on each of 4 plates. Top with a dollop of the apple chutney and a sprig of fresh thyme.
Heat oil in a medium saucepan over medium-high heat. Add the onion, ginger, garlic, and jalapeno, and cook until soft. Add half of the apples, the apple juice and vinegar, and cook until the apples are soft. Remove from the heat and stir in the remaining apples and the thyme. Season with salt and pepper, to taste. Transfer to a bowl and let sit at room temperature for 30 minutes before serving so that flavors meld.
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