1. Heat a grill or grill pan to medium (if using a charcoal grill, coals are ready when you can hold your hand 5 inches above grill for just 6 to 8 seconds). In a food processor, puree one-quarter of the garlic (about 1/2 teaspoon), 1 tablespoon thyme, the cannellini beans, and 1 teaspoon oil until smooth. If mixture is too thick to spread, add a bit of water. Season with pepper, and pulse to combine.

2. Combine the remaining garlic and remaining 1 tablespoon thyme with 1 tablespoon plus 2 teaspoons oil in a shallow dish. Place the mushroom caps in the garlic mixture, and turn to coat. Season with pepper; drizzle with vinegar.

3. Place onion slices on a plate, and lightly coat each side with remaining 2 teaspoons oil. Grill the onions until lightly charred on the first side, about 3 minutes. Flip the onions, and continue grilling until tender and charred on the other side, about 3 minutes more. Transfer onions to a clean plate; cover to keep warm.

4. Working in batches if necessary, place mushrooms on the grill, stem side up. Grill until browned on the first side and juices have begun to collect in the centers, about 5 minutes. Flip mushrooms, and continue cooking until stem side of each cap is browned and center is tender, about 4 minutes more.

5. Split hamburger buns, and place cut side down on grill; cook just until warm and toasted. Spread 1/4 cup bean puree on the bottom half of each bun, and top with 2 grilled mushroom caps. Layer each with sliced cheese, grilled onions, and a small handful of arugula. Top with remaining roll halves, and serve.