Fill a medium saucepan halfway with water. Bring to a boil and add 1 tablespoon salt. Add the chicken, reduce heat to medium, and gently simmer until cooked through, 12 to 14 minutes. Transfer the chicken to a cutting board and let rest at least 5 minutes before slicing.
Meanwhile, in a small bowl, whisk together the oil, lime juice, honey, 1/2 teaspoon salt, and 1/4 teaspoon pepper.
Divide the arugula among bowls and top with the chicken, avocado, and radishes. Drizzle with the vinaigrette. Serve with the bread (if using).