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Return cooking liquid to a boil and cook spaghetti, stirring occasionally, 5 minutes (pasta will not be fully cooked). Reserve 1 cup of pasta water and drain pasta in colander and return empty pot to stovetop. Add wine and sugar to pot and boil vigorously 2 minutes until liquid is reduced a bit. Add spaghetti and shake pot to prevent pasta from sticking. Gently stir with tongs until coated and boil over high heat, stirring occasionally, until most of the liquid is absorbed, about 6 minutes (pasta will be al dente). Immediately after adding spaghetti to wine mixture, cook garlic and red pepper flakes in the olive oil in a large, deep skillet over moderately low heat, shaking skillet occasionally, until garlic is pale golden, about 5 minutes. Add broccoli, salt, and pepper and cook, stirring, 1 minute. Add 1/2 cup of reserved pasta water.
Pour broccoli into skillet with the spaghetti mixture and carefully toss with tongs to combine (skillet will be very full). Cook while stirring, about 2 minutes. Remove from heat, season with freshly ground black pepper and drizzle with a bit of olive oil. Stir in the grated cheese. Serve immediately.