Heat 1 teaspoon of olive oil in a small frypan over medium heat. Cook and stir the onions in the oil until soft and translucent, about 5 minutes.
Gently knead together the softened onions, lamb, egg, bread crumbs, garlic, salt and pepper. Divide the mixture into 6 parts and roll into balls then cover and refrigerate until ready to use.
Mix together the goats cheese, extra-virgin olive oil, basil and oregano until well combined; cover and chill for 5 minutes.
Preheat the barbecue for medium-high heat.
Working with one portion of meat at a time make an indentation in the centre of the ball with your thumb. Fill the indentation with a heaping tablespoon of the goats cheese mixture. Gently pull and form the meat patty around the cheese filling, making a burger-shaped patty. Repeat with each ball of the lamb mixture.
Barbecue the patties until no longer pink in the centre and well done, about 8 minutes per side.