Take a 20cm x 30cm (8-inch x 12-inch) (top measurement) lamington pan and grease all sides. Line base and long sides with baking paper, extending the paper 5cm (2 inches) over the edges.
Process flour, icing sugar and butter until crumbly. Add 1 egg yolk and enough of the iced water until the ingredients cling together. Knead dough on a floured surface until smooth. Press dough into base of pan; prick all over with a fork. Cover pan with plastic wrap; refrigerate for 30 minutes.
Preheat oven to 200C/400F. Remove plastic from pan and bake the base for 15 minutes or until browned lightly.
Combine milk and cream in a medium saucepan. Split vanilla bean lengthways; scrape seeds into pan with bean. Bring cream mixture almost to the boil. Whisk remaining egg yolks, caster sugar and arrowroot in a small bowl until combined. Place gelatine and the boiling water in a small bowl; stir to dissolve. Discard bean from cream mixture; gradually whisk hot cream mixture into egg mixture. Return to pan; cook, whisking, until custard boils and thickens. Whisk gelatine mixture through custard.
Spread custard over base; cool to room temperature. Top with strawberries; brush with jam. Refrigerate for 1 hour, before cutting into 15 slices.