Portobello mushrooms are a nutritious and satisfying alternative to burgers made with beef or turkey. We topped these burgers with smoked cheddar cheese, but you can use another type of cheese or omit it entirely.
Heat a grill or grill pan over medium heat. Make bean puree: In a food processor, combine one-quarter of garlic (about 1/2 teaspoon), 1 tablespoon thyme, the cannellini beans, and 1 teaspoon oil; process to form a smooth and spreadable paste. If mixture is too thick, add 1 teaspoon water. Season with pepper, and pulse to combine. Set aside.
Combine the remaining garlic, tablespoon thyme, and oil on a plate or in a baking pan. Place the mushroom caps in the garlic mixture, and turn to coat. Season with pepper; drizzle with vinegar.
Place onion slices on a plate, and lightly coat each side with cooking spray. 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; keep warm.
Place mushrooms on the grill, stem side up, working in batches if necessary. 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.
Split hamburger buns in half, and place cut side down on the grill; cook until warm and toasted. Transfer to a work surface. Spread 1/4 cup bean puree on the bottom half of each bun, and top with two grilled mushroom caps. Layer each with sliced cheese, grilled onions, and a small handful of arugula. Top each with matching roll halves, and serve.
To clean the mushroom caps, wipe them with a damp paper towel; do not rinse or soak them because they will remain soggy even after cooking.