Make the Egg Salad:
Place eggs in a small saucepan and cover with cold water by 1 inch. Set over medium high heat and bring to a boil. As soon as the water is boiling, remove the pan from the heat, cover and let sit for 10 minutes.
Remove eggs from the water with a slotted spoon and transfer to a bowl filled with cold water and ice. Let the eggs rest in the ice bath while you prepare the remaining ingredients.
Combine the remaining ingredients in a medium bowl. Peel the eggs and cut in half. Transfer the yolks to the bowl. Whisk all ingredients together until smooth. Cut the egg whites into a medium dice. Fold into the mayonnaise mixture. Season to taste with salt and pepper.
Transfer rye bread slices to a clean surface. Butter the bread. Place a handful of arugula leaves on top of the butter. Divide egg salad evenly between the slices of bread.
Top each sandwich with radish slices if desired. Serve.
Calories: 240kcal | Carbohydrates: 17g | Protein: 9g | Fat: 15g | Saturated Fat: 4g | Cholesterol: 172mg | Sodium: 421mg | Potassium: 153mg | Fiber: 2g | Sugar: 2g | Vitamin A: 431IU | Vitamin C: 3mg | Calcium: 55mg | Iron: 2mg