1/2 pineapple, halved lengthwise, cored, cut crosswise into 1/4-inch-thick slices
2 mangoes, peeled, halved, cut crosswise into 1/4-inch-thick slices
1/4 cup apricot preserves, melted
Directions
Blend first 4 ingredients in processor until nuts are finely ground. Using on/off turns, cut in butter until pea-size pieces form. Add yolk, 1 teaspoon water, and extract; using on/off turns, blend just until moist clumps form, adding more water by teaspoonfuls if dough is dry. Gather into ball; flatten into disk. Wrap in plastic and chill 3 hours.
Roll out dough between 2 lightly floured sheets of waxed paper to 1/4-inch thickness. Peel off top sheet of paper. Invert crust into 11-inch round tart pan with removable bottom. Peel off paper; press dough into pan. Trim edges. Push crust 1/8 inch above top edge of pan. Refrigerate 1 hour.
Preheat oven to 375°F. Bake crust until golden, about 25 minutes. Cool. (Can be made 1 day ahead. Wrap and store at room temperature.)
Combine milk and cream in heavy medium saucepan over medium heat. Scrape in seeds from vanilla bean; add bean. Bring to simmer. Whisk yolks, sugar, and cornstarch in large bowl to blend. Gradually whisk hot milk mixture into yolk mixture; return to saucepan and whisk over medium heat until thick and beginning to bubble, about 8 minutes. Transfer to medium bowl. Cool slightly; mix in rum. Press plastic wrap onto surface of custard. Chill until cold, at least 4 hours. (Can be made 1 day ahead. Keep chilled.)
Remove vanilla bean from filling. Whisk filling just until smooth; spread in crust. Arrange pineapple and mango slices in concentric circles atop filling. Brush fruit with preserves. (Tart can be assembled 3 hours ahead. Refrigerate.)
