Melt the butter in your skillet over medium-low heat, then add the sliced onions with a pinch of salt. Stir occasionally and let them cook low and slow for about 25 to 30 minutes. You're waiting for them to turn a deep golden brown, not just soft and translucent. Resist the urge to rush this with high heat, since that just leads to burnt edges and raw centers.
While the onions cook, season your chicken breasts generously with salt and pepper. Push the onions to one side of the pan, add the olive oil, and sear the chicken for about 3 to 4 minutes per side until golden. It doesn't need to cook through yet, since it will finish in the oven later. Remove the chicken to a plate and set it aside for a few minutes.
Add the garlic and thyme to the onions and cook for about a minute until fragrant. Sprinkle the flour over the onions and stir it in to cook off the raw taste. Pour in the wine to deglaze the pan, scraping up any browned bits stuck to the bottom, then add the beef broth and Worcestershire sauce. Let this simmer for a few minutes until it thickens slightly into a glossy sauce.
Nestle the seared chicken breasts back into the onion sauce, spooning some of the onions over the top of each piece. Preheat your oven to 400°F while you top each breast generously with the shredded cheese. Transfer the whole skillet into the oven and bake for 15 to 18 minutes, or until the chicken reaches 165°F internally and the cheese is bubbly and golden.
Pull the skillet out and let it rest for about 5 minutes before serving. This gives the juices time to redistribute through the chicken instead of running out onto the plate. Scatter the fresh parsley over the top right before serving for a pop of color. That short resting period genuinely makes a difference in how juicy each bite turns out.