First, heat vegetable oil and butter in a large pan over medium-low heat until the butter is melted. Add the onion and stir to coat, then cook until the onions begin to turn translucent, about 10 minutes. Then, stir in the sugar and salt and cook until the onions are softened, about 10 more minutes. Next, add the sliced green apple, water, and balsamic vinegar and stir to combine. Cook until the apples are softened and the onions are caramelized, about 15 to 20 minutes.
Next, slice the baguette both lengthwise and crosswise into squares or rectangles. Drizzle the cut side of each piece of bread with extra virgin olive oil, then broil on LOW until lightly toasted, about 3 to 4 minutes.
Meanwhile, slice the brie into strips, starting from the center of the wheel and working outwards. Then remove the rind from each piece of cheese. Make sure to size the slices of brie to fit each piece of bread, so that the whole surface of each sandwich can be covered in a single layer of cheese.
Once the caramelized onions and apples are done, top each piece of toasted bread with a single layer of brie and broil on LOW until brie is melted, about 2 minutes. Then top each sandwich with a serving of caramelized onions and apples. Serve hot.