Step 1

Thinly slice 2 pounds yellow onions. Melt 2 tablespoons unsalted butter in a 10-inch or larger cast iron skillet over medium-high heat. Add the onions, 1/8 teaspoon baking soda, and 1/2 teaspoon kosher salt. Cook, stirring regularly with a wooden spoon, for 15 minutes. Continue cooking, stirring more frequently and scraping up the browned buildup that forms on the bottom of the pan and reducing the heat as needed, until the onions are a deep golden-brown, about 10 minutes more. Transfer the onions to one side of a large plate; set aside.

Step 2

Meanwhile, if using chicken breasts, slice each one in half horizontally, starting at the thick end, so that you have 4 (1/4-inch-thick) cutlets total. Pat the chicken dry and season all over with the remaining 1 teaspoon kosher salt and 1/2 teaspoon black pepper.

Step 3

Place an oven rack 6 inches below the broiler and turn the oven to broil.

Step 4

Heat 2 tablespoons olive oil in the same skillet over medium-high heat until shimmering. Add the chicken, in batches if needed, and cook, flipping once, until lightly browned and cooked through, 3 to 4 minutes per side. Transfer the chicken to the plate with the onions, tent with aluminum foil, and repeat with any remaining pieces of chicken. Meanwhile, grate 2 ounces Gruyère cheese on the largest holes of a box grater; set aside.

Step 5

Add 1 cup beef or chicken broth and 1 tablespoon Dijon mustard to the pan and stir to combine, scraping up the browned bits from the bottom of the pan. Bring to a boil. Lower the heat to maintain a simmer and cook until reduced by half, 1 to 2 minutes. Meanwhile, coarsely chop 2 tablespoons fresh parsley leaves.

Step 6

Return the chicken to the skillet, top with the caramelized onions, and sprinkle with the Gruyère. Broil until the cheese is melted, about 2 minutes. Sprinkle with the parsley and serve immediately.