Make the potatoes: Cover potatoes with cold water and season with enough salt so that the water tastes like the sea. Bring to a simmer and cook until potatoes are soft, approximately 45 minutes. Drain and cool on a baking sheet (do not run them under cool water). Set aside at room temperature.
Make the peperonata: In a large pot, heat ½ cup of the olive oil over medium heat until shimmering. Add garlic and cook, stirring, until they just start to turn golden, 3 to 4 minutes. Stir in onions, increase heat to medium-high, and cook for 2 minutes. Stir in peppers and cook, stirring occasionally, until they start to soften, about 10 minutes. Add tomatoes and basil, and stir to combine. Bring to a gentle simmer, then reduce heat to maintain simmer. Continue to cook, stirring occasionally, until peppers are very soft, about 30 minutes. Stir in the remaining olive oil and vinegar, and season with salt and pepper to taste.
Make the octopus: To a large saucepan over low heat, place celery, onions, and bay leaves in bottom of pan; add garlic on top. Nestle octopus on top of vegetables. Add white wine, cover, and let the octopus stew in its own juices. Uncover after 1 hour to check progress. There should be a lot of liquid in the pan. To test for doneness, stick a fork in the thickest part of octopus; it’s done if the fork slides out easily. Cover and continue to stew over low heat until the octopus is tender, about 20 minutes more, depending on the size of the octopus. Transfer octopus to a baking sheet to cool. Once cool, cut octopus: To remove tentacles from body, flip octopus upside down. You will then see where all eight tentacles meet. Following the natural lines, carve with a sharp knife at the base of every tentacle, trying to keep them all as long as possible. Season with blackening spice and a light sprinkle of salt. Next, place on grill or grill pan on medium heat and cook for 2 minutes on each side.
Make the salsa: In a blender, puree parsley, basil, mint, olive oil, capers, anchovies, garlic, lemon juice and zest, until herbs are finely blended, scraping down sides of blender as necessary. Season with salt and pepper to taste, and transfer to a small bowl. Use immediately.
To assemble: Spread warm peperonata on serving platter. Arrange octopus and potatoes over top. Liberally dress the dish with salsa verde and serve.