Heat oven to 375 degrees F. Lightly butter a 10-inch tart tin with a removable bottom. Set aside. Lightly dust a clean work surface with flour, and roll dough into a circle with a 14-inch diameter. Fit circle into prepared tin being careful not to stretch dough too thin. Transfer to freezer until firm, about 15 minutes. In a food processor fitted with the steel blade, finely grind almonds. Set aside. Make the frangipane: combine the 1/2 cup of butter and sugar in bowl of an electric mixer fitted with paddle attachment. Beat until light and fluffy, about 2 minutes. Add ground almonds, 3 tablespoons rum, egg, almond extract, and flour, and beat until smooth. Spread evenly into chilled tart shell and refrigerate while preparing pears. Remove cooled poached pears from liquid and cut each in half lengthwise, removing core and stem. Place each half, cut side down, on a cutting board and cut crosswise into thin slices. Arrange sliced pear halves on frangipane around edge of tart, leaving space between each half, and place 1 half in center of tart. When arranging pears, try to "pull" slices toward center of tart, which will elongate pears a bit and fill shell better than if pears were just placed flat. Bake until the tart shell is golden brown and the frangipane is puffed and brown, 40 to 45 minutes. While the tart is baking, melt apricot jam with remaining 1 tablespoon rum in a small saucepan over medium heat. Brush warm-from- the-oven pears with apricot-rum glaze. Let cool tart to room temperature before slicing.
Combine the flour, sugar, and salt in the bowl of a food processor fitted with the steel blade. Pulse to combine. Add butter and process, just until the mixture resembles a coarse meal about 10 seconds With machine running add 1/4 cup ice water, in a slow steady stream through feed tube. Drizzle in egg yolk and process just until the dough holds together without being wet or sticky; about 30 seconds. Test dough at this point by squeezing a small amount together. If it is crumbly, add a bit more water. Divide dough in half. Pat into discs and wrap in plastic wrap. Chill at least 1 hour. Yield: 2 (8 to 10-inch) tarts or single crust pies
Combine wine, 4 cups water, the sugar, lemon juice, lemon zest, cinnamon, and vanilla bean in a large saucepan. Bring to a boil and cook for 5 minutes. Add pears, reduce to simmer, and cook until the pears are tender when pierced with the tip of a knife, 20 to 30 minutes. Cool pears in poaching liquid. Yield: 4 servings
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