Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until fork-tender, about 20 minutes. Drain, cool, and cut into quarters.
Stir potatoes, diced egg whites, whole egg yolk, celery, green onions, and minced radishes together in a large bowl. Add mayonnaise, pickle relish, mustard, chives, mustard-mayonnaise sandwich sauce, parsley, kosher salt, basil, and white pepper; carefully stir until well-combined and coated. Cover the bowl with plastic wrap and refrigerate until flavors blend, 8 hours to overnight.
Garnish potato salad with radish roses and egg slices.