Chicken Stuffed with Brie and Balsamic Caramelized Onions

2 large onions, cut in half lengthwise and thinly sliced into half moons
1 tbsp balsamic vinegar
4 skinless, boneless chicken breasts, trimmed of fat
4 oz Brie, sliced or cut into chunks
1 cup flour
1 cup dry bread crumbs or panko
4 tbsp grated Parmesan
2 tbsp  chopped fresh parsley (optional)
Salt and pepper
2 eggs

Thinly slice the onions. Put the oil in a large heavy bottomed pan and heat over medium high heat and add the onions. Stir and season with salt. Turn down to medium heat. Let caramelize for about 20 minutes, stirring occasionally (frequently if you don't have a heavy bottomed pan.) Once the onions are golden and caramel colored (but not burnt), pour in the balsamic, stirring to pick up the sugars on the bottom of the pan. Let cook for about a minute, then remove from the heat.

Preheat the oven to 400 degrees.

Place the chicken on a cutting board, and cut a slit horizontally along one edge of the breast, cutting nearly to the opposite side but not all the way through. Open it so it forms two flaps, attached at the center, like a butterfly. Stuff each breast with about a quarter of the caramelized onions and a quarter of the Brie; close the flap, press it down firmly, and set them aside on a baking sheet. Don’t worry that they aren’t sealed around the edges.

In a shallow dish, combine the bread crumbs, Parmesan, parsley, and salt and pepper to taste. In another dish, lightly beat the eggs with a fork, and in a third dish place the flour.  Holding a chicken breast together, dip it in the flour, then the beaten egg, coating it on both sides, then in the bread crumb mixture, turning it over to coat it well. Repeat with each chicken breast.

Drizzle a little oil into a large ovenproof skillet set over medium-high heat. Cook the chicken breasts until browned on one side, then turn them over and place the skillet in the oven. If you don’t have an ovenproof skillet, brown the chicken breasts on the other side, then transfer them to a baking sheet or dish and put them in the oven.  Bake the chicken for 15-20 minutes, until no longer pink inside. (Poke them with a sharp knife to peek if you can’t tell.) Serve immediately.