For the octopus: Place cleaned octopus and flakey salt into a stand mixer with the hook attachment. Beat the octopus on the lowest speed until the tentacles curl and can be separated, 15 to 20 minutes; this tenderizes the meat. Remove the octopus from the mixer, rinse under cold running water in a colander, drain and pat dry. Cover and refrigerate until ready for use. For the vinaigrette: Combine and whisk together the vinegar, garlic, Simple Preserved Lemons, Meyer lemon zest and juice in a medium glass mixing bowl. Slowly add and whisk in the oil until completely incorporated and smooth. Add the basil and gently whisk. Season with salt and pepper. Cover and refrigerate until ready for use. For the salad: Preheat the grill to high heat. Separate the tentacles and head of the octopus. Skewer each leg, weaving the legs through the skewers. Using tongs and an oil-blotted paper towel, wipe down the grill grates to clean and create a nonstick surface. Drizzle the octopus with a few tablespoons of olive oil and sprinkle with kosher salt and pepper. Place the octopus onto the hot grill and sear. Allow to char around the edges then turn, 3 to 4 minutes per side. The octopus is done when just cooked through yet nicely charred around the edges. Do not overcook or it will be tough. Remove and let rest for 1 to 2 minutes. Cut the octopus tentacles and head into bite-size pieces, and then toss with about 3 tablespoons of the lemon vinaigrette. Set aside. Toss the arugula, white beans, chile, radishes and red onions with the remaining vinaigrette. Add the warm octopus and carefully toss to combine. Season with flaky sea salt and pepper. Serve immediately, garnishing with more preserved lemon, if desired, and a drizzle of extra-virgin olive oil.
Quarter the lemons and rub with the salt. Combine in a mason or canning jar and press the lemons to release juice. Top with water to cover. Seal and age for up to 2 weeks in the refrigerator.
To clean fresh octopus, turn the head inside out and remove the ink sac, innards and mouthparts. Rinse gently and pat dry.