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Bring 4 quarts water to boil in large pot. Pat chicken dry with paper towels. Heat oil in large skillet over medium-high heat until just smoking. Cook chicken until no longer pink, 1 to 2 minutes per side. Transfer to plate.
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Add pasta to boiling water and cook until just beginning to soften, about 8 minutes. Add asparagus to pot and cook until bright green and pasta is al dente, about 4 minutes.
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Reserve 1 cup cooking water, drain pasta and asparagus, and return to pot.
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Whisk peanut butter, vinegar, oyster sauce, chili garlic sauce, ginger, sesame oil, and 1/2 cup pasta cooking water in bowl until smooth.
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Add peanut butter mixture and chicken to pot and toss to combine, adding reserved pasta water as needed. Serve.