Preheat grill to high heat and scrape grates clean. Turn all burners to low heat.
In a food processor, add the fresh basil, walnuts, Parmesan cheese, garlic, olive oil, lemon juice and lemon zest. Pulse until a smooth paste forms.
Place the onions on a large sheet of grilling foil. Drizzle with about one tablespoon of the lemon pesto. Fold to seal the foil and place the packet of onions on the grill.
In a large bowl, mix together the ground turkey, salt and remaining lemon pesto. Form into four burgers.
Place the lemon pesto turkey burgers on the grill. Flip the burgers after about 5 minutes. Continue grilling the burgers until the internal temperature reaches at least 165 degrees Fahrenheit.
Remove the foil packet of onions from the grill.
Serve the burgers on the toasted brioche buns with Boston lettuce, goat cheese and the grilled onions.
