1 large Vidalia or other sweet onion (1 to 1 1/4 lb, at least 4 inches in diameter)
8 very thin (1/4-inch) slices firm white bread, crusts discarded
1/3 cup mayonnaise
1/2 cup finely chopped fresh parsley

Cut 4 (1/8-inch-thick) slices from widest part of onion, keeping rings intact.

Spread each slice of bread with about 1 teaspoon mayonnaise and top 4 slices with 1 onion slice each. Top with remaining bread slices, mayonnaise sides down, then trim onion flush with bread. Quarter sandwiches into small square or triangular sandwiches.

Arrange 2 sheets of wax paper on a work surface. Spread remaining 2 1/2 tablespoons mayonnaise in a thin layer on 1 sheet and spread parsley evenly on the other.

Dip all edges of 1 sandwich in mayonnaise to coat, then dip in parsley. Transfer to a tray and repeat with remaining sandwiches.

Cooks' note:
Sandwiches can be made 2 hours ahead. Wrap tray with plastic wrap and chill.