Reserve 6 large garlic cloves. Put remaining cloves in 2 cups cold water, then simmer in a small saucepan, covered, until garlic is very soft, about 30 minutes. Reserve 1/2 cup garlic cooking liquid, then drain garlic in a sieve.
Puree cooked garlic with reserved cooking liquid and 1 teaspoon salt in a blender until smooth.
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). Transfer to a bowl and cover tightly with plastic wrap, then let stand 20 minutes. Peel peppers, discarding stems and seeds, and cut into 3/4-inch pieces.
Mince 2 reserved garlic cloves with thyme and remaining teaspoon salt using a large knife. Transfer along with spinach to a large serving bowl.
Cook pasta in a large pot of boiling salted water until al dente.
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. Add peppers and cook, stirring, 3 minutes. Add beans and garlic puree to skillet and bring to a simmer over moderately low heat, stirring occasionally. Season with salt and pepper.
Reserve 1 cup cooking water, then drain pasta. Add pasta, sauce, vinegar, and cheese to spinach and garlic in serving bowl and toss to combine. (Add some of reserved cooking water if mixture is dry.) Serve immediately.