To prepare the chicken, whisk olive oil, balsamic vinegar, and salt and pepper, to taste, in a medium bowl. Slice horizontally through each chicken breast, starting at the thick end. Add chicken to the bowl and toss to combine. Set aside until ready to grill.

To prepare the onions, trim off both ends, cut them in half, remove skins, and slice into strips from root to stem end. Melt butter over medium heat in a large skillet. Add the onions to the pan, stirring to coat with butter. Continue cooking over medium heat, stirring every 5 to 10 minutes, scraping up burned bits from the bottom of the pan as needed. After about 20 minutes, lower the heat if the onions seem to be burning in places. At about 40 minutes, the onions should be golden and very soft. If they taste great at this point, you can add remaining ingredients and finish cooking. If you like a deeper flavor, continue cooking until you're satisfied. Once the onions reach the desired flavor and texture, add 1/4 cup balsamic vinegar, scraping up the burned bits on the bottom as you combine it with the onions. Add a bit of water, if needed, to get all the goodness off the bottom of the pan. Sprinkle in salt, to taste. Remove from heat and use immediately or store in a sealed container for up to a week in the refrigerator.

Heat grill to high heat. Add chicken, turning every 2 to 3 minutes until cooked through, about 6 to 10 minutes.

To assemble sandwiches, spread a generous layer of chutney on the bottom half of each ciabatta roll. Add chicken, slicing as needed if using smaller rolls, and top it with Brie, balsamic onions, and arugula. Season with salt and pepper and top with the remaining halves of each roll.