Place the first 6 ingredients in a mason jar. Cover tightly with lid and shake vigorously until well combined.
Season to taste with salt and pepper, and freshly grated nutmeg.
Vinaigrette will last up to 1 week in the refrigerator. Let sit at room temperature 15 minutes before using.
Place radish slices in a large bowl and pour in enough vinaigrette to coat without soaking. Gently toss. Let marinate 10-15 minutes. Set aside. Reserve remaining dressing for when serving or another use.
Meanwhile, toast the walnuts if they’re not already, and prep the remaining ingredients.
After radishes marinate, arrange them on a platter. Sprinkle the feta, walnuts, shallots, pomegranate arils, and fresh herbs over the radish slices.
Garnish with extra parsley, if desired. Serve with extra vinaigrette on the side. Enjoy!