Combine mustard and sugar in a small bowl. Stir in 1 1/2 tablespoons water. Add soy sauce and 1 teaspoon oil. Set aside.
Heat a grill or grill pan until very hot. Brush salmon with remaining oil; season with salt and pepper. Grill for 5 minutes, or until browned. Turn and cook for another 5 minutes, or to desired doneness.
Toast bread and place on plates. Top each with wilted arugula and a salmon fillet. Drizzle mustard sauce over top. Serve immediately.
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