This dish is all about the vinaigrette. Extra limey and rich from the avocado, it makes a simple summertime dish that will impress any guest. Good olive oil and a lot of freshly picked basil are important, so don’t skimp on these items.
Preheat the grill to high heat (400˚F to 450˚ F).
Arrange the onion slices on a baking sheet and brush lightly with the olive oil. Season the onions lightly with 1/2 teaspoon of the salt and 1/2 teaspoon of the pepper. Grill the onion slices for 4 to 5 minutes on each side or until tender, sweet and slightly charred. Transfer the grilled slices to a mixing bowl and separate the rings. Add the tomato and avocado slices to the onion rings and set aside.
In a separate bowl, whisk together the lime juice, extra-virgin olive oil and chopped basil. Toss the vinaigrette with the grilled onion mixture and season the salad with 1 teaspoon of the salt and 1/2 teaspoon of the pepper. Allow the salad to marinate at room temperature for 10 to 15 minutes, tossing occasionally. (This allows the avocados to break down a little and slightly thicken the vinaigrette.)
Season the fish filets on both sides with the remaining 11/2 teaspoons of salt and 11/2 teaspoons of pepper. Place the filets on the hot grill and cook for 6 to 7 minutes or until the filets begin to pull away and loosen from the grill rack. Turn and cook an additional 4 to 5 minutes or until the filets are golden brown and cooked through. Remove the filets from the heat and keep warm until ready to serve.
Arrange 1 marinated tomato slice in the center of each of 6 dinner plates. Using a slotted spoon, place onion and avocado slices on top of the tomatoes. Top each salad with a second marinated tomato slice. Spoon several tablespoons of the vinaigrette over and around each salad. Arrange the grilled grouper filets on top of each salad. Toss the arugula in the mixing bowl with the remaining vinaigrette. Top each fish filet with a small mound of the arugula. Serve immediately.