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FOR MERINGUE: Position rack in center of oven and preheat to 350 degrees F.
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Spread 1 1/2 cups coconut on large baking sheet.
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Toast until golden, stirring twice, about 15 minutes.
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Cool.
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Maintain oven temperature.
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Line another large baking sheet with foil.
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Using electric mixer, beat egg whites and next 4 ingredients in large bowl until foamy.
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Gradually add sugar and beat until stiff peaks form.
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Beat in 1/4 cup boiling water, 1 T at a time, beating until whites are stiff and glossy.
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10
Fold in 1 cup toasted coconut.
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Spoon meringue onto center of prepared baking sheet and spread to 9-inch-diameter circle with slightly raised edges.
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Sprinkle with remaining 1/2 cup untoasted coconut.
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Bake meringue for 10 minutes.
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14
Reduce oven temperature to 200 degrees F and bake until dry and crisp outside and just cooked through inside, about 1 hour.
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15
Turn off oven.
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Let meringue stand in oven 1 hour.
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Remove from oven and cool completely.
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FOR MOUSSE: Whisk egg yolks, 2 T sugar, butter, 2 T water, and rum in medium metal bowl to blend.
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Set bowl over saucepan of simmering water (do not allow bottom of bowl to touch water).
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Whisk until thermometer inserted into mixture registers 160F, about 6 minutes.
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Add chocolate and whisk until melted and smooth.
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Turn off heat; leave bowl over water.
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Using electric mixer, beat egg whites in large bowl until soft peaks form.
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Gradually add 2 T sugar and beat until stiff peaks form.
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Fold 1/3 of egg whites into chocolate mixture to lighten.
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Fold in remaining whites.
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Remove from over water and let mousse cool 20 minutes.
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28
Beat whipping cream in large bowl until peaks form.
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Place meringue on platter.
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Spread mousse over center of meringue; top with sliced bananas.
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31
Spread whipped cream over bananas and chill at least 20 minutes and up to 3 hours.
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Sprinkle with remaining toasted coconut.
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Cut into wedges and serve.