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.
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.
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.
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.
Set chops on plates and spoon onion mixture over them.
Calories 603 (54% from fat); Fat 36g (sat 10g); Protein 43g; Carb 28g; Fiber 1g; Chol 133mg; Sodium N/A.