For the burgers:
1) Prepare a medium-hot fire in a charcoal barbecue with a cover, or preheat a gas griddle to medium-high.
2) Combine the beef, basil, sun-dried tomatoes, onion, garlic, and salt in a large bowl, handling as little as possible. Shape into 6 patties to fit the bun size. Loosely cover with cling film and set aside.
3) When the barbecue is ready, brush the griddle rack with vegetable oil. Place the patties on the griddle, cover, and cook, turning once, until done to preference, 5 to 7 minutes on each side for medium.
4) Place the cheese slices on the patties during the last 3 minutes of griddling. Place the buns, cut side down, on the outer edges of the rack to toast lightly during the last 2 minutes of griddling.
5) To assemble the burgers, place equal portions of the griddled fennel on each bun bottom, followed by a cheese-topped patty, 2 bacon slices, and an equal portion of the fig-rocket topping. Add the bun tops and serve.
For the fennel:
1) Grate 2 tsps zest from the lemons. Juice the lemons to yield 30ml of juice. Put the fennel rings in a medium-sized bowl and toss with the lemon juice, oil, and salt. Put the fennel in a griddle basket and griddle, shaking the basket occasionally, until soft, 10 to 12 minutes.
2) Transfer the fennel to a sheet of foil, sprinkle with the lemon zest, and wrap to keep warm.
3) Heat a large, heavy nonstick fire-proof frying pan. Add the bacon and cook until crisp. Transfer to paper towels to drain. Wrap in foil to keep warm.
For the rocket-fig topping:
1) Heat a large, heavy nonstick fire-proof frying pan. Add the walnuts to the pan and toast until golden and fragrant; set aside.
2) Whisk the vinegar with the oil in a small bowl and season with salt, to taste. Combine the figs, rocket, and toasted walnuts in a medium bowl. Toss with just enough dressing to coat.