Add 8 tablespoons of butter to a heavy-bottomed pot over low heat, and once melted, add the sliced onions and sea salt. Stir to combine, place a lid on top, and let the onions wilt for 30 minutes.
After 30 minutes, take the lid off of the skillet and toss the onions in the butter. Turn the heat up to medium-low, and let the onions continue to caramelize, uncovered, for 45 minutes, scraping the pot and stirring the onions occasionally. Once caramelized, remove the onions from the pot, and set aside.
Preheat the oven to 375°F. Dust the chicken breasts with flour, salt, and pepper.
Turn the heat to medium-high, add 3 tablespoons of butter to the pot, and once melted, sear the chicken breasts for 3-4 minutes on each side, until a golden brown crust is formed. Remove the chicken from the pot and set aside.
Add the caramelized onions, red wine, and chicken broth to the pot, stirring to combine. Let the onion mixture simmer for 5-6 minutes, until slightly thickened and reduced. Then, nestle each chicken breast into the caramelized onion mixture, making sure to cover each piece of chicken with the onions. Top each piece of chicken with shredded gruyere cheese and a sprig of fresh thyme.
Bake for 10 minutes. Then let cool slightly, serve, and enjoy!