1. Rinse pork chops. In a 1 1/2- to 2-quart pan over medium heat, stir beer, salt, 2 tablespoons brown sugar, the peppercorns, mustard seeds, fennel seeds, and garlic until salt and sugar are dissolved. Let mixture cool completely. Place chops in a 2-gallon zip-lock plastic bag and add brine. Seal and chill 4 to 6 hours.
2. Remove chops from brine (discard brine). Rinse chops and pat dry. Pour olive oil into a 12-inch frying pan over medium-high heat. When hot, add chops and cook, turning once, until well browned on both sides, about 6 minutes total.
3. Transfer chops to a plate. Add onions to pan. Stir frequently until onions are very soft and browned, 20 to 25 minutes (lower heat if necessary to prevent scorching). Stir in vinegar and remaining 2 tablespoons brown sugar and cook, scraping bottom of pan to loosen any browned bits, until liquid is almost completely evaporated.
4. Place chops on top of onion mixture and cover pan; cook until chops are barely pink in the center (cut to test), 6 to 8 minutes.
5. Set chops on plates and spoon onion mixture over them.
Calories 603 (54% from fat); Fat 36g (sat 10g); Protein 43g; Carb 28g; Fiber 3.1g; Chol 133mg; Sodium N/A.