Stir together 1/3 cup mayonnaise and next 8 ingredients. Spread remaining 1/3 cup mayonnaise evenly over 1 side of each bread slice. Spread 4 bread slices, mayonnaise side up, evenly with half of egg salad. Top evenly with half of spinach and 4 bread slices.
Repeat procedure with remaining egg salad, spinach, and bread slices. Cut each sandwich into quarters; garnish, if desired.