Prepare a water bath by filling a large bowl with ice and a small amount of water. Bring a large pot of salted water to a boil. Blanch each bean variety separately in the boiling water 2 to 3 minutes or until tender (cooking times may vary depending on the size and variety of bean). Plunge the beans into the ice water bath and stir until cool. Drain and pat the beans dry. Cut any larger beans to match the sizes of the smaller beans. Set the beans aside.
Whisk together the lemon juice, olive oil, shallots, basil, parsley, chives, and thyme in a medium bowl. Season the vinaigrette with ¼ teaspoon each of salt and pepper. Set the vinaigrette aside.
Salt and pepper the cobia. Place the cobia on a hot grill and cook 4 to 5 minutes on each side or until the fish is cooked through and flakes easily when tested with a fork.
Combine the blanched beans, cherry tomatoes, frisée, watercress, and arugula in a large bowl with the vinaigrette. Season the salad with the remaining ¼ teaspoon of salt and ¼ teaspoon of pepper. Divide the salad evenly among 6 dinner plates and place 1 piece of cobia on each salad. Drizzle 2 tablespoons of the aioli over each plate. Serve immediately.