Step 1

Arrange a rack in the middle of the oven and heat to 325°F. Cook the onions and chicken while the oven is heating.

Step 2

Melt the butter in a large, heavy bottomed oven-safe pot or Dutch oven over medium-high heat. Add the onions and baking soda and cook, stirring every few minutes with a wooden spoon, for 15 minutes. Continue cooking, stirring more frequently and scraping up the browned buildup that forms at the bottom of the pan, until the onions are a deep golden-brown, 10 to 15 minutes more. Meanwhile, sear the chicken.

Step 3

Heat the oil in a large nonstick or cast iron pan over medium-high heat until shimmering. Meanwhile, pat the chicken dry with paper towels and season with salt and pepper. Working in batches as to not overcrowd the pan, sear the chicken until golden brown on both sides, about 3 minutes per side. As each batch is ready, transfer the chicken to a large plate.

Step 4

Add 1 cup of the broth to the chicken pan and scrape up any browned bits at the bottom of the pan. Add the mustard and vinegar and whisk to combine, then simmer until reduced by about half, about 5 minutes. Remove from the heat.

Step 5

When the onions are ready, add the garlic, thyme, and rosemary, and cook until fragrant, about 1 minute. Add the remaining 1 cup of broth, scrape up any browned bits at the bottom of the pan, and simmer until reduced by half, about 5 minutes. Taste and season with salt and pepper as needed. Remove and discard the thyme and rosemary stems.

Step 6

Transfer the chicken on top of the onions in an even layer and add any juices on the plate. Pour the mustard sauce over the chicken. Bake uncovered for 30 minutes.

Step 7

Remove the pot from the oven, then turn the oven up to broil. Sprinkle the chicken evenly with the cheese. Broil until the cheese is melted and golden brown on top, 3 to 4 minutes. Sprinkle with the parsley if desired and serve.