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1
Heat a grill or grill pan over medium heat.
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2
Make the bean puree: In a food processor, combine one-quarter of the garlic (about 1/2 teaspoon), 1 tablespoon thyme, the cannellini beans, and 1 teaspoon oil; process to form a smooth and spreadable paste.
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3
If the mixture is too thick, add 1 teaspoon water.
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4
Season with pepper, and pulse to combine.
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5
Set aside.
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6
Combine the remaining garlic, tablespoon thyme, and oil on a plate or in a baking pan.
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7
Place the mushroom caps in the garlic mixture, and turn to coat.
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8
Season with pepper; drizzle with vinegar.
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9
Place the onion slices on a plate, and lightly coat each side with cooking spray.
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10
Grill the onions until lightly charred on the first side, about 3 minutes.
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11
Flip the onions, and continue grilling until tender and charred on the other side, about 3 minutes more.
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12
Transfer to a clean plate; keep warm.
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13
Place the mushrooms on the grill, stem side up, working in batches if necessary.
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14
Grill until browned on the first side and juices have begun to collect in the centers, about 5 minutes.
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15
Flip the mushrooms, and continue cooking until the stem side of each cap is browned and center is tender, about 4 minutes more.
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16
Split the hamburger buns in half, and place cut side down on the grill; cook until warm and toasted.
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17
Transfer to a work surface.
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18
Spread 1/4 cup bean puree on the bottom half of each bun, and top with 2 grilled mushroom caps.
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19
Layer each with sliced cheese, grilled onions, and a small handful of arugula.
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20
Top each with matching roll halves, and serve.
