Preheat the oven to 375 degrees F, ensuring one rack is positioned in the center of the oven & a second rack is positioned directly under the broiler. Gather & prep all ingredients according to ingredients list.
Add 2 tablespoons of the butter to a 12-inch oven-safe skillet over medium heat. Season the chicken with S&P. Once the butter is melted, place the chicken in the skillet, browning for 2-3 minutes per side. Transfer the seared chicken to a plate (it will not be cooked through at this point – totally fine & expected!) & set aside.
If needed, add another 1-2 tablespoons unsalted butter to the same skillet used to sear the chicken. Reduce heat to medium-low. Once the butter is melted, add the sliced onions, stirring to coat. Cook, stirring occasionally, for 10-12 minutes until softened & browned. (The onions shouldn’t require too much attention at this point!)
Once softened, season the onions with 1/2 teaspoon kosher salt, stirring to combine. Continue to cook, stirring occasionally, until until tender, fragrant, & deeply browned, 20-30 minutes more. Be sure to adjust heat as needed to allow the onions to caramelize slowly & prevent burning. The closer the onions get to reaching caramelization, the more often you will want to stir to promote even browning & prevent burning.
Once caramelized, deglaze the skillet. Slowly pour in the sherry wine, using a wooden spoon to scrape up any browned bits formed on the bottom of the pan. Increase the heat to medium-high to bring the sherry to a simmer, then simmer 4-5 minutes or until reduced completely. Add the flour, Dijon mustard, & thyme to the skillet, stirring to coat the onions, then add in the beef stock. Bring to a simmer.
Nestle the browned chicken from Step 2 into the caramelized onion sauce. Spoon some of the onion sauce over top the chicken breasts, then sprinkle the grated cheese over top. Transfer the skillet to the center rack of the preheated oven. Cook for 15-20 minutes, or until an instant-read thermometer inserted in the thickest portion of the chicken registers an internal temperature of 165 degrees F.
For a toasty golden brown finish, transfer the skillet to the oven rack under the broiler & turn the broiler on high. Broil 2-3 minutes or until the cheese is as browned as you’d like – be sure to keep a close eye on the chicken as it broils to prevent burning.