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1
Combine water, 1 cup white sugar, honey, vanilla bean, cinnamon stick, and cardamom in a large saucepan. Bring to a simmer over medium heat. Add pears; reduce heat and simmer until pears are translucent or easily pierced with a knife, about 10 minutes. Remove from heat and cool to room temperature.
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2
Combine 1 1/2 cup flour, confectioners' sugar, and salt in a bowl. Cut in 1/2 cup plus 1 tablespoon butter until mixture is crumbly. Mix in egg yolk until dough sticks together in large clumps.
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3
Grease a 9-inch tart pan with a removable bottom. Press dough evenly into the pan. Freeze for 30 minutes. Reserve leftover dough for patching cracks.
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4
Preheat oven to 375 degrees F (190 degrees C).
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5
Grease the shiny side of a piece of aluminum foil with butter; place butter-side down over the pastry. Transfer tart pan to a baking sheet.
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6
Bake in the preheated oven, using leftover dough to patch any cracks, until dough looks dry and light brown, about 20 minutes. Peel off aluminum foil; let pastry cool.
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7
Reduce oven temperature to 350 degrees F (175 degrees C).
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8
Combine 2/3 cup white sugar and 1/3 cup butter in a large bowl; beat with an electric mixer until light and fluffy. Mix in ground almonds. Add egg, egg yolk, 2 teaspoons flour, and cornstarch; beat until frangipane is smooth. Beat in almond extract and vanilla extract.
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9
Spread frangipane evenly over the pastry. Drain pear halves and dry with paper towels. Cut lengthwise into even slices. Place 1 pear half on a spatula and gently fan out slices. Place on frangipane. Repeat with remaining pear halves.
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10
Bake in the preheated oven until frangipane is firm to the touch, 40 to 45 minutes. Cool on a wire rack.
