Preheat the oven to 450°F.
Toss the potatoes and carrots on a baking sheet with the oil, salt, and pepper. Roast until fork-tender, but not mushy, about 15-18 minutes.
Combine all of the pesto ingredients except the oil in a blender, food processor, or immersion blender. Process until mostly smooth. With the motor running, slowly add the oil until you reach a smooth and creamy consistency. It is important that you add the oil slowly and not all at once. Season to taste.
Preheat an indoor or outdoor grill to medium.
Season the salmon fillets with salt and pepper. Place the fish on the grill skin side up for 5 minutes. Flip and grill for another 5 minutes. The internal temperature of the salmon will be 115°F at medium rare. If you prefer your salmon more well done, cook it for another 1-2 minutes per side.
To serve, divide the salmon, potatoes and carrots between plates. Top with pesto.