Heat the oven to 425°F. Lightly grease or line an 18-by-13-inch rimmed sheet pan. Pat the salmon dry.
Toss the potatoes on the pan with 1 Tbsp olive oil, ½ tsp salt, and ¼ tsp pepper. Arrange them cut-side down and roast for 18 minutes.
Whisk the lemon zest, 2 Tbsp lemon juice, 3 Tbsp olive oil, garlic, Dijon mustard, parsley, dill, ¾ tsp salt, and ¼ tsp pepper. Divide the mixture between two bowls.
Turn the potatoes. Add the green beans and tomatoes, tossing them with half of the lemon-herb mixture.
Clear four spaces on the pan. Add the salmon skin-side down and brush it with the remaining lemon-herb mixture.
Roast for 12–15 minutes, until the salmon separates into moist flakes and reaches 145°F at its thickest point. Make sure the potatoes are fork-tender.
Rest the salmon for 5 minutes, then serve it with the roasted vegetables and pan juices.