To make the octopus: Coat a large saucepan with olive oil and toss in the garlic, carrot, celery and onion. Season with salt and crushed red pepper and bring the pan to a medium-high heat. Cook the veggies until they are soft and aromatic, 7 to 8 minutes. Add the vinegar and fill the pan with water. Season with salt, toss in the bay leaves and bring the water to a boil. Reduce to a simmer and simmer for 20 minutes. Add the baby octopus and simmer for 20 minutes. Remove from the liquid and let cool. (This part can sooooooo be done ahead.)
To prep the salad: Coat a large straight-sided saute pan with olive oil, toss in the onions, season with salt and crushed red pepper. Bring the pan to a medium heat and cook the onions until they start to soften, 2 to 3 minutes. Add the garlic and cook for 1 to 2 minutes. Toss in the beans, chicken stock and celery and cook until the stock has reduced by a little more than half. Turn off the heat and reserve the bean mixture.
Preheat the grill. Cut the heads off the octopus and cut the legs into sections of 2 legs. Oil the legs and place them on the grill to slightly char and heat up the octopus.
In a large bowl, toss the arugula and celery leaves with the lemon juice and olive oil and season with salt. Divide among 4 serving plates and spoon some of the warm bean mixture onto the arugula. Arrange the legs between the plates. Drizzle with big fat finishing oil.
Cooked Cannellini Beans:
Drain the beans from the soaking water and place in a medium saucepan. Toss in the garlic cloves, bay leaf, carrot, celery, thyme and onion. Add water to the pan until it covers the beans by about 2 inches.
Put the pan on a high heat and bring the water to a boil, reduce to a simmer and simmer until the beans are very soft and tender, 35 to 40 minutes. Turn the heat off.
Season the water generously with salt and let sit for 15 minutes. Drain the beans from the cooking liquid and remove the veggies, bay and thyme and discard. Yield: 2 1/2 cups cooked beans.