1. Mash eggs with mayonnaise and next 4 ingredients until well blended. Season with salt and pepper to taste. Cover and chill up to 1 day.
2. Toss together arugula, radishes, lemon juice, and olive oil. Season with salt and pepper to taste. Toss to coat.
3. Spread egg salad mixture on 1 side of each bread slice; top 7 slices with arugula mixture. Top with remaining 7 bread slices, egg salad side down. Trim crusts from sandwiches; cut each sandwich into 3 rectangles with a serrated knife.