Peel and core the pears using a melon baller or paring knife, keeping the neck intact. Combine the wine, water, sugar, vanilla bean, and lemon juice in a large shallow pot over medium heat and bring to a boil. 2.Add only as many pears as can float and move freely, reduce to a simmer, and poach, stirring occasionally, until a knife meets little resistance. Test several pieces, as they cook at different speeds. Drain and cool the pears and poaching liquid separately, then store the pears in the cooled liquid until needed.

Mix the flour, salt, and sugar together, then cut in the cold butter until pieces are approximately ⅛ inch (dried lentil size). 4.Make a well in the center, add cold water gradually, and mix just until a soft, shaggy dough forms. Pat into a round, wrap, and refrigerate for at least 30 minutes.

Beat the butter in a stand mixer fitted with the paddle attachment until soft and smooth. Add the almond paste and beat on low until the lumps break down, then on high until smooth. Add the egg slowly in several additions, beating well after each. 6.Add the cornstarch and mix until no dry streaks remain, then add the rum and mix to combine. Refrigerate until needed, up to 7 days.

Preheat the oven to 350°F (177°C). Let the dough rest at room temperature for 10 to 20 minutes until malleable. Roll into a 10-inch disk, approximately ⅛ inch thick, on a lightly floured surface. Chill until firm, then line the tart ring or removable-bottom tart pan, chill for 10 to 15 minutes, and dock the shell.

Briefly paddle the frangipane until light and fluffy, then spread an even layer in the shell using an offset spatula. Slice each poached pear half crosswise into ¼-inch slices, keeping the shape intact. Arrange the five pear halves with their narrow tops toward the center and fan them inward. Scatter sliced almonds over any exposed frangipane.

Bake for 40 to 50 minutes, until the crust is deeply browned, the pear edges are caramelized, and the frangipane is puffed, golden, and springs back slightly when pressed in the center. Brush the pears with warmed apricot nappage and let set before slicing.