Combine the cloves, carrots, garlic, shallot, wine and 2 cups water in a medium Dutch oven or sauce pan. Place over medium-high heat and bring to a simmer.
Meanwhile, rinse the octopus. Remove the head just above the eyes and then make a cut just below the eyes to remove them completely. Dip the ends of the octopus tentacles in the simmering wine mixture for a few seconds. Remove and then dip them again. Repeat once more for a total of 3 times; this will curl the tips for a very pretty result.
Submerge the octopus tentacles and the head in the wine mixture, adding more water if needed to cover the octopus. Reduce the heat to low and simmer gently, covered, until the tip of a paring knife goes into the thickest part without much resistance, about 1 hour 45 minutes. Allow the octopus to cool for 30 minutes in the liquid before transferring to a plate to cool completely. When cool enough to handle, use a piece of paper towel to wipe the skin away. Cut the base into individual legs.
Heat a grill pan over medium-high heat. Drizzle the octopus legs and head with 1 tablespoon of the oil and sprinkle with 1/4 teaspoon salt. Place on the hot grill and cook until charred in a few spots, about 4 minutes per side. Set aside to cool.
Preheat the oven to 400 degrees F.
Place the bread cubes, Parmesan, 4 tablespoons oil and 1/4 teaspoon salt in a medium bowl. Mix well to coat. Scatter the cubes on a baking sheet and bake until golden brown and crispy, about 12 minutes. Add the toasted cubes back to the bowl along with the arugula, tomatoes and olives. Tear in the basil. Drizzle the lemon juice and the remaining 1/3 cup oil around the edge of the bowl and toss well to marry all the ingredients. Chop the octopus into bite-size pieces and add them to the salad along with 1/8 teaspoon salt. Toss one more time and serve.
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