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1
Reserve 6 large garlic cloves.
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2
Put remaining cloves in 2 cups cold water, then simmer in a small saucepan, covered, until garlic is very soft, about 30 minutes.
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3
Reserve 1/2 cup garlic cooking liquid, then drain garlic in a sieve.
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4
Puree cooked garlic with reserved cooking liquid and 1 teaspoon salt in a blender until smooth.
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5
While garlic simmers, roast peppers on racks of gas burners over high heat, turning with tongs, until skins are blackened, 10 to 12 minutes (or broil peppers on a broiler pan about 5 inches from heat, turning occasionally, about 15 minutes).
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6
Transfer to a bowl and cover tightly with plastic wrap, then let stand 20 minutes.
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7
Peel peppers, discarding stems and seeds, and cut into 3/4-inch pieces.
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8
Mince 2 reserved garlic cloves with thyme and remaining teaspoon salt using a large knife.
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9
Transfer along with spinach to a large serving bowl.
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10
Cook pasta in a large pot of boiling salted water until al dente.
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11
Make sauce while pasta is cooking: Finely chop remaining 4 garlic cloves and cook in oil in a 12-inch nonstick skillet over moderate heat, stirring, until pale golden, 3 to 5 minutes.
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12
Add peppers and cook, stirring, 3 minutes.
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13
Add beans and garlic puree to skillet and bring to a simmer over moderately low heat, stirring occasionally.
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14
Season with salt and pepper.
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15
Reserve 1 cup cooking water, then drain pasta.
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16
Add pasta, sauce, vinegar, and cheese to spinach and garlic in serving bowl and toss to combine.
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17
(Add some of reserved cooking water if mixture is dry.)
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18
Serve immediately.
