Spread the Brussels sprouts on a small baking sheet. Drizzle with 2 tablespoons of olive oil and season with ½ teaspoon salt and ½ teaspoon pepper. Toss to coat.
Bake for 45–50 minutes, or until the sprouts are browned and crispy.
While the Brussels are roasting, marinate the salmon: In a small bowl, combine 1 tablespoon of olive oil, ⅛ teaspoon salt, ⅛ teaspoon pepper, and the brown mustard.
Place the salmon, skin-side down, in a glass baking dish and pour the mustard mixture over it. Spread the mixture evenly over the salmon, then marinate at room temperature for 20 minutes.
Just before the salmon is finished marinating, prepare the arugula salad: In a medium bowl, toss the arugula with the lemon juice, 1 tablespoon of olive oil, and ¼ teaspoon salt, or to taste.
Heat the remaining tablespoon of olive oil in a medium pan over medium heat. Pan fry the salmon for about 4 minutes on each side, or until cooked to your desired level of doneness. Transfer to a serving plate.
Sprinkle the Brussels sprouts with the mustard seed powder and plate with the salmon and arugula. Squeeze a bit of lemon juice over the salmon.