For the fish: Gently rub the green onions over the Moi, then season with salt and pepper. In a large, hot saute pan, add the olive oil and gently place the fish in the pan. Sear over medium heat for 10 to 15 minutes, turning over 2 to 3 times, until crisp and golden brown.
For the Lomilomi: In a hot saucepan, add the olive oil and heat. When the olive oil is hot, add the onions and tomatoes and lightly sweat until the onions are soft and translucent. Next, add in shrimp, garlic, ferns, and ginger and continue to saute. Deglaze with the stock and reduce by half. Reduce the heat and whisk in the butter until melted. Finish with the lemon juice and green onions. Season, to taste, with salt and pepper.
Place the arugula in the center of a plate. Place the fish on top of the arugula and top off with the sauce.
In a large, hot saute pan, add the olive oil and arugula. Toss lightly and add in the garlic and saute for 1 minute. Season, to taste, with salt and pepper.