In a large skillet, bring 2 inches water to a simmer. Line the inside of a steamer basket with the lettuce leaf and basil. Season arctic char with salt and pepper on both sides; arrange on top of the lettuce and basil in the basket. Line a second basket with another lettuce leaf and sprig of basil. Top with beans.
When the water is simmering, set the basket with the beans in the skillet (make sure the water level does not flood the bottom basket), and top with the basket holding the arctic char. Secure a lid tightly in place on top. Steam until beans are just tender and brightly colored, and fish is evenly opaque and cooked throughout, about 7 to 9 minutes. If the beans are not yet tender, remove char, cover beans with the lid, and return to skillet to cook for 1 to 2 minutes more.
Meanwhile, make the vinaigrette: Whisk together mustard, lemon juice, sugar, garlic, and basil. Slowly add oil in a steady stream and whisk continuously until emulsified. Season to taste with salt and pepper.
To serve, drizzle vinaigrette over fish and beans. Garnish with small basil leaves.