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1
Bring 4 quarts water to rolling boil, covered, in stockpot.
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2
Meanwhile, heat 1 tablespoon butter in 12-inch nonstick skillet over high heat until just beginning to brown, about 1 minute. Add chicken in single layer; cook for 1 minute without stirring, then stir chicken and continue to cook until most, but not all, of pink color has disappeared and chicken is lightly browned around the edges, about 2 minutes longer. Transfer chicken to clean bowl; set aside.
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3
Return skillet to high heat and add 1 tablespoon butter; add onion and 1/4 teaspoon salt and cook, stirring occasionally, until browned about edges, 2 to 3 minutes. Stir in garlic, red pepper flakes, thyme, and flour; cook, stirring constantly, until fragrant, about 30 seconds. Add wine and chicken broth; bring to simmer, then reduce heat to medium and continue to simmer, stirring occasionally, until sauce has thickened slightly and reduced to 1 1/4 cups, about 15 minutes.
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4
While sauce simmers, add 1 tablespoon salt and broccoli to boiling water; cook until broccoli is tender but still crisp at center, about 2 minutes. Using slotted spoon, transfer broccoli to large paper towel-lined plate. Return water to boil; stir in pasta and cook until al dente. Drain, reserving 1/2 cup pasta cooking water; return pasta to pot.
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5
Stir remaining 2 tablespoons butter, Asiago, sun-dried tomatoes, parsley, and chicken into sauce in skillet; cook until chicken is hot and cooked through, about 1 minute. Off heat, season to taste with pepper. Pour chicken/sauce mixture over pasta and add broccoli; toss gently to combine, adding pasta cooking water as needed to adjust sauce consistency. Serve immediately, passing additional Asiago and the lemon wedges (if using) separately.
