"Above and beyond the ingredients, the cooking method, or should I say methods, couldn't be easier. As long as you braise it gently on low heat, until just tender, and then give it a nice sear before serving, you should be in great shape. This smoky braised octopus is great served with crusty roasted potatoes."
Place chopped onion, garlic cloves, bay leaf, paprika, salt, and 1 tablespoon olive oil into saucepan. Cook and stir over medium-high heat until onions start to soften, about 5 minutes. Add white wine to create the braising liquid. Bring to a simmer.
Transfer octopus into braising liquid, turning to coat with the liquid. Cover; reduce heat to low. Simmer 20 minutes and turn octopus over. Continue cooking until octopus is tender, another 40 to 45 minutes.
Remove from heat; transfer octopus to a large bowl. Pour braising liquid over octopus. Place bowl above a bowl of ice water to cool to room temperature.
Cover bowl with plastic wrap when octopus and liquid are cool. Refrigerate until thoroughly chilled, at least 2 hours or overnight.
Transfer octopus to a work surface and mop off braising juices with paper towels; remove as much of the purple skin as you want. Cut octopus into 3 or 4 serving-size pieces. Brush with olive oil.
Transfer braising liquid to saucepan. Bring to a boil; remove from heat. Strain out solids; transfer liquid to a mixing bowl; let cool about 10 minutes. Stir in lemon juice, extra-virgin olive oil, parsley, salt, and cayenne pepper to make the serving sauce.
Heat a grill pan or outdoor grill to high heat. Transfer octopus pieces to grill. Cook until flesh is seared, almost charred, 3 or 4 minutes. Turn and cook other side until dark brown, caramelized, and heated through, another 3 or 4 minutes. Transfer to a platter.
Slice octopus into fairly thick diagonal slices. Place on a serving dish. Spoon braising liquid mixture over the top. Dust with a bit of cayenne pepper, if desired.