Preheat and Prepare: Position an oven rack in the center and preheat your oven to 425°F (220°C). Line a large baking sheet (a 12x17 inch sheet works well) with parchment paper.
Dress the Vegetables: In a medium bowl, combine the carrot sticks and green bean halves. In a separate small bowl, whisk together the olive oil, 1 ½ tablespoons of the lemon juice, ⅜ teaspoon of the salt, the garlic powder, and the optional red pepper flakes. Pour this dressing over the vegetables and toss gently to coat them evenly. Spread the seasoned vegetables in a single layer on one side of the prepared baking sheet.
First Roast for Vegetables: Place the baking sheet into the preheated oven and roast the vegetables for 10-12 minutes, until they just begin to soften.
Add the Salmon: Carefully remove the baking sheet from the oven. Gently stir the vegetables or push them to maintain their side. Arrange the salmon fillets, skin-side down (if using skin-on), on the empty side of the baking sheet.
Season the Salmon: Drizzle the remaining 1 ½ tablespoons of lemon juice over the salmon fillets. Sprinkle them evenly with the remaining ⅜ teaspoon of salt, the ½ teaspoon of black pepper, and the ½ teaspoon of dried dill weed.
Final Roasting: Reduce the oven temperature to 400°F (200°C). Return the baking sheet (now with both vegetables and salmon) to the oven. Bake for an additional 10-15 minutes. The exact time will depend on the thickness of your salmon fillets and your preferred level of doneness. Salmon is cooked when it flakes easily with a fork or reaches an internal temperature of 135-145°F (57-63°C).
Serve Hot: Once cooked, remove the baking sheet from the oven. Serve the Lemon Pepper Salmon and roasted vegetables immediately.