Prepare a medium-hot fire in a charcoal grill with a cover, or preheat a gas grill to medium-high.

For the patties:

Combine the chuck, 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 plastic wrap and set aside.

For the fennel:

Grate 2 teaspoons zest from the lemons. Juice the lemons to yield 2 tablespoons juice. Put the fennel rings in a medium-sized bowl and toss with the lemon juice, oil, and salt. Put the fennel in a grill basket and grill, shaking the basket occasionally, until soft, 10 to 12 minutes. Transfer the fennel to a sheet of foil, sprinkle with the lemon zest, and wrap to keep warm.

Heat a large, heavy nonstick fire-proof skillet on the grill. Add the bacon and cook until crisp. Transfer to paper towels to drain. Wrap in foil to keep warm.

For the topping:

Drain off the bacon fat from the skillet, wipe out the skillet with paper towels, and set the skillet back on the grill. Add the walnuts to the skillet and toast until golden and fragrant; set aside.

Whisk the vinegar with the oil in a small bowl and season with salt, to taste. Combine the figs, arugula, and toasted walnuts in a medium bowl. Toss with just enough dressing to coat.

When the grill is ready, brush the grill rack with vegetable oil. Place the patties on the rack, cover, and cook, turning once, until done to preference, 5 to 7 minutes on each side for medium. Place the cheese slices on the patties during the last 3 minutes of grilling. Place the buns, cut side down, on the outer edges of the rack to toast lightly during the last 2 minutes of grilling.

To assemble the burgers, place equal portions of the warm grilled fennel on each bun bottom, followed by a cheese-topped patty, 2 bacon slices, and an equal portion of the fig-arugula topping. Add the bun tops and serve.