Cook the olive oil, sesame oil, red onion, garlic, ginger, and red pepper flakes in a small, heavy-bottomed pot on medium heat until the onion is transparent, 10 to 15 minutes. Whisk in the vinegar, sugar, soy sauce, peanut butter, ketchup, sherry, and lime juice; cook for 1 more minute. Cool.
Using a sharp knife, carefully remove the entire breast of the rotisserie chicken and slice it diagonally in 1/2-inch-thick slices. Skewer the chicken slices with wooden sticks and serve with the sauce. Reserve the remaining chicken.
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