Place the whole octopus in a pot of salted water with the beer and bay leaf and bring to a boil.
Let boil for 30 to 45 minutes, until fork tender.
Drain and rinse the octopus under cold water and let it cool completely in the refrigerator. This helps firm up the meat, making it easier to slice thinly.
Lay out a piece of plastic wrap on a cutting board.
Place the cooled octopus on the plastic wrap and then cover it with another piece of plastic wrap. Use a rolling pin or heavy pan to gently flatten the octopus to an even thickness.
Preheat the grill to 375˚F (190˚C) and lightly oil the grates.
Grill the octopus for 2 to 3 minutes on each side, seasoning with salt and pepper.
Once grilled, place the octopus in the refrigerator for up to an hour.
In a bowl, combine the Dijon, lemon zest, lemon juice, cilantro, and a drizzle of olive oil. Season with salt and pepper to taste.
Take the octopus out of the refrigerator and, using a sharp knife, slice the octopus as thinly as possible lengthwise. Shingle the thinly sliced octopus on a platter.
Toss the arugula lightly in half of the dressing and place on top of the octopus.
Drizzle the remaining vinaigrette over the octopus. Serve immediately.