Place the salmon, pesto, egg white, breadcrumbs, shallot, and salt and pepper in a food processor and whiz until the salmon is finely chopped, but not puréed. Chill in the refrigerator for 30 minutes.
Divide the mixture into 8 evenly sized salmon burgers.
Heat a little olive oil in a large nonstick frying pan over medium heat. Add the burgers and cook