For the turkey mince, sauté ground turkey in a pan with olive oil, salt, cumin, and pepper for about 5-8 minutes until completely cooked through. Set aside.
Create pesto: lightly toast walnuts in a dry skillet for 3 minutes or until just starting to become fragrant. Blitz together all ingredients for pesto except olive oil in a food processor, then slowly stream in the olive oil until a coarse paste is created.
Toast sourdough over a lightly oiled skillet or in the oven. Assemble sandwiches with onion, tomato, arugula, turkey mince, and a drizzle of pesto.