Generously sprinkle all sides of the pork loin with salt and pepper. Place a braiser or shallow Dutch oven over medium-high heat. Add 2 tablespoons of the oil and heat until shimmering. Carefully put the pork in the pan. Sear until deeply golden on all sides, 1 to 2 minutes per side. Remove from the pan.
Put the onions and apple in a large mixing bowl. Add the caraway seeds, dried thyme, remaining 1 tablespoon oil, 1 teaspoon salt and 1/2 teaspoon pepper and toss to coat. Transfer the mixture to the bottom of the braiser and nestle in the garlic cloves. Pour in 1/4 cup of the chicken stock and place the pork back on top.
Roast until the pork reaches an internal temperature of 145 degrees F, about 30 minutes. Transfer the pork to a cutting board to rest while you make a pan sauce. Remove the apple and onion mixture to a platter
Return the braiser to the stove over medium-high heat. Pour in the hard cider and remaining 1/2 cup chicken stock. Cook, scraping with a spatula to remove any brown bits from the bottom of the pan. Whisk in the mustard. Allow the sauce to simmer until reduced slightly, a couple of minutes. Add the butter, whisking until melted. Cook just until the sauce is shiny and slightly thickened, about 5 minutes. Taste and season with more salt and pepper if needed.
Slice the pork and arrange the slices over the onion and apple mixture. Top with sauce and chopped parsley.