Preheat the oven to 425°F (220°C), move the rack to the top shelf.
Make pesto: wash basil and gently pat dry the leaves. Add all the ingredients for the pesto except olive oil to a food processor. Process until almost smooth. Add the olive oil (add just 2 tablespoons right now) and gently pulse until just combined (don't mix it at high speed). Assess if the sauce needs another tablespoon of olive oil (if the basil was not drained properly, the water can dilute the sauce making the third olive oil tablespoon not necessary, for this recipe you need the pesto to be rather thick and not runny). Season to taste with salt and pepper.
Pat dry salmon fillets with paper towels, season with salt and pepper on all sides.
Spread 1 tablespoon of pesto on each fillet.
In a small bowl, combine breadcrumbs, parmesan cheese, and olive oil. Spread the mixture on top of each salmon fillet.
Bake for about 12 minutes or until a thermometer registers 140°F/60°C (after the resting time it should come to the safe temperature of 145°F/63°C). Leave to rest for 5 minutes then serve.
Enjoy!