Cut each chicken breast crosswise into thin strips. Heat a large skillet over medium to medium-high heat. Add 1 Tbsp. oil, and when it is hot, sauté the chicken, turning occasionally, until it is browned and cooked through, about 5 minutes. (Set aside some cooked chicken, sliced nectarines, and cucumber for non-salad eaters, if necessary.)
In a large serving bowl, combine the chicken, nectarines, cucumbers, almonds, and scallions.
In a small bowl, whisk together 2 Tbsp. oil, the vinegar, lime juice, honey, curry powder, ginger, and salt. (Alternatively, you can shake them all up in a jar to emulsify them.)
Pour the dressing over the salad and toss well. Gently toss in the mint, if desired. Refrigerate it for at least 15 minutes and up to 2 days. Season the salad with salt and pepper, to taste, at the table.