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 to a cutting board and let rest for at least 5 minutes before slicing.
Meanwhile, in a small bowl, whisk together the oil, lime juice, honey, ½ teaspoon salt, and ¼ teaspoon pepper.
Divide the arugula among bowls and top with the chicken, avocado, and radishes. Drizzle with the vinaigrette. Serve with the bread (if desired).