Combine first 3 ingredients in a large bowl, whisking well. Stir in edamame and next 4 ingredients. Set aside.
Heat 2 tablespoons butter in a skillet over medium-high heat. Cook 2 trout fillets, skin side down, 5 minutes or until skin becomes crispy and lightly browned. Turn fish and cook an additional 5 minutes or until done. Repeat with remaining butter and trout.
Spoon edamame salad on each of four plates. Top with trout and Crab-stuffed Piquillos. Garnish, if desired.