Heat milk and ginger in large saucepan on medium heat 5 min., stirring occasionally, until small bubbles form around sides of saucepan. Meanwhile, in heatproof bowl, whisk egg yolks with sugar and cornstarch until smooth. Gradually whisk about 1/4 of the hot milk mixture into yolk mixture. Then whisk contents of bowl into saucepan. Cook over medium-low heat, stirring continuously, 10 to 12 min. until custard is thickened. Remove from heat. Stir in cold butter and vanilla. Immediately strain through a sieve into a bowl (discard any lumps); cool custard completely. (Can be made ahead and kept refrigerated up to 2 days in airtight container.)
To make tart, preheat oven to 425°(220°C). Place pastry on parchment-paper lined baking sheet; score 1 in. (2.5 cm) border on all sides of pastry with sharp knife. Prick (dock) pastry all over with fork or tip of sharp knife. Bake 17 to 18 min. or until golden and flaky. (After pastry is baked, cool completely; can be made ahead and kept at room temperature up to 6 hr.)
Meanwhile, line parchment-paper lined baking sheet with softened butter and sprinkle with brown sugar. Lay bananas, cut-side down, on butter and brown sugar. Roast alongside pastry, 7 to 8 min or until softened and flat sides are golden brown.
Whisk custard to loosen. Spread inside score lines on pastry. Arrange roasted bananas, cut-side up, on top. Garnish with lime zest.
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