Set the top oven rack to the center of your oven, and preheat to 325 F. Pat chicken completely dry, and season with 1 teaspoon of salt and 1/2 teaspoon of black pepper on both sides. Press to adhere.
In a large, oven safe skillet (I use a large cast iron), add 1 tablespoon of vegetable/avocado oil over medium high heat. Once shimmering, add the seasoned chicken. Sear on each side until the flesh has browned, about 4-5 minutes on each side. Place seared chicken on a plate and set aside.
Reduce the heat of your pan to medium and add the remaining tablespoon of oil, along with the garlic and shallot. Stir frequently, until the garlic becomes fragrant, about 1 minute. Add the caraway seeds and cook for an additional minute.
Add the broth and wine, and bring the liquid to a simmer. Use the back of your spoon to scrape up any brown bits along the bottom of the pan.
Nestle the chicken back into the pan, along with any juices that have accumulated on your plate. Place pan in preheated oven and braise uncovered until your chicken breast reaches an internal temp of 165 F at the thickest part, about 38-42 minutes.
Remove pan from the oven (using oven mitts!) to the stovetop. Use tongs to place chicken on a plate and tent with foil. Over high heat, bring the sauce to an aggressive simmer and continue to cook until the sauce has reduced to about 1 1/2 a cup of liquid, about 8-10 minutes.
Whisk the water and cornstarch together in a small bowl, then pour into the sauce. Remove your pan from heat, and stir in the crème fraîche and dill. Taste the sauce and adjust seasoning to your liking. Return chicken to the pan, and drizzle with sauce. Enjoy!