Bring a large pot of lightly salted water to a boil; cook rotini at a boil until tender yet firm to the bite, about 8 minutes. Drain and rinse with cold water.

Mix pasta with chicken, green pepper, and green onions in a large bowl.

Mix mayonnaise, brown sugar, vinegar, lemon juice, and salt together in a small bowl until well blended. Pour dressing over the pasta mixture and stir well to coat.

Cover and refrigerate until flavors are combined and dressing is thickened, 8 hours to overnight. Stir in cashews just before serving, or no more than 2 hours ahead.