To tenderize the fresh or frozen octopus, place it in a pot and cover with water. You can season the water if you wish with salt, pepper, garlic, bay leaves, pickling spice, etc. Bring the water to a boil and cook the octopus for 40 to an hour depending on the size. If using precooked octopus, skip this step.
Remove the octopus from the hot water, rinse and place into a bowl. Drizzle with the olive oil and chopped garlic and let it cool and rest at room temperature for 30 minutes to an hour.
Preheat your grill or broiler to medium-high heat and then clean and oil the grates. Working over direct high heat sear4s and crisps the octopus without the risk of drying out the interior.
Place the octopus on the preheated grill or under the broiler and sear until browned and crisped on one side, about 4 minutes. Carefully flip the octopus and brown the other side, about 4 minutes longer. NOTE: Grilling times may vary depending on the heat of your grill / broiler and the distance the octopus is to the heat source.
When done, remove the octopus from the grill and transfer to a plate or platter. Serve as is with the herb-oil dipping sauces or slice the octopus into smaller pieces and serve with the sauces as is or on top of a bed of greens or on a salad. Garnish with lemon wedges if desired.