Rinse the octopus under cold water. Cut off the octopus body (that bulbous sac over the eyes) set aside. Remove the next part of the body -- the part with the eyes and discard, leaving just the tentacles. Flip the tentacles upside down and press the beak out of the opening at the center of the octopus. Discard the beak.
Place the octopus body and tentacles in a large pan or dutch oven. Add the carrots, celery, onion, crushed garlic, parsley, wine, olive oil and wine corks. Cover the pot tightly with a lid and heat to boiling. Reduce heat to simmer and cook for 45 minutes to an hour, until octopus is firm but tender. Remove from heat. Place octopus on a platter and let cool.
When octopus is cool enough to handle, run your fingers over the body and tentacles and slough off the skin, it will come off in patches - be careful not to work the tentacles too hard, or they will break off.
Place body and tentacles in a large bowl. Add sliced garlic, red pepper flakes, lemon zest, parsley leaves and olive oil. Using tongs or your hands, toss to coat the octopus. Cover with plastic wrap and refrigerate 6 hours or overnight.
Heat the grill to high heat, about 450 degrees. Sear the octopus on the grill, weighting it down with a heavy pan to achieve the best crust. Cook for 2-3 minutes and remove from grill. Cut off the tentacles and slice the body into 1/2-3/4" thick rounds.
Arrange octopus on a platter. Drizzle with olive oil and squeeze of lemon. Sprinkle with red onions and flat leaf parsley. Serve with additional lemon wedges.