Combine butter, powdered sugar, and egg white in a medium bowl, stirring until well blended. Mix in flour and cocoa powder, stirring until a soft dough forms. Press dough into a 10-inch tart pan. Cover and chill 1 hour.
Line crust with parchment paper, and fill with pie weights or dried beans. Bake at 375° for 35 minutes. Remove parchment and weights; let crust cool.
Heat 1 1/4 cups cream in a heavy saucepan over medium heat until very hot. Remove from heat, and add 12 ounces chocolate, stirring until smooth. Stir in eggs and sugar, whisking until blended. Strain to remove lumps, if necessary.
Fill crust with chocolate mixture. Bake at 300° for 1 hour 15 minutes. Cool completely.
Combine remaining 1/4 cup cream and remaining 4 ounces chocolate in a glass measuring cup. Microwave on high at 30-second intervals, stirring occasionally, until mixture is smooth. Spread over top of cooled tart. Let stand 30 minutes or until set. Slice and serve with Hawaiian Vanilla Sauce and fresh berries.