In a small saucepan, bring . cup (120 grams) water to a boil over medium-high heat. Remove from heat, and add basil leaves. Let basil steep for 5 minutes. Strain basil juice, discarding solids. Refrigerate until chilled.

In the work bowl of a food processor, place flour, sugar, and salt; pulse until combined. Add cold butter, pulsing until mixture is crumbly. Add ? cup plus 2 teaspoons (90 grams) ice water, basil, and basil juice, pulsing until dough just comes together. Divide dough in half, and shape each half into a disk. Wrap tightly in plastic wrap, and refrigerate for 30 minutes.

Preheat oven to 350°F (180°C).

On a lightly floured surface, roll half of dough into a 12-inch circle. Transfer to a 9-inch pie plate, pressing into bottom and up sides. Trim excess dough to . inch beyond edge of plate. Fold edges under. Refrigerate for at least 30 minutes.

Top dough with a piece of parchment paper, letting ends extend over edges of plate. Add pie weights.

Bake for 15 minutes. Carefully remove paper and weights. Bake until golden brown, about 10 minutes more. Let cool for 10 minutes.

Fold mangoes into Frangipane, and pour mixture into prepared crust.

On a lightly floured surface, roll remaining dough into a 9.-inch circle. Using a sharp knife, cut .-inch-wide strips. Reserve 3 strips. Arrange remaining strips over filling in a lattice design.

Brush water on rim of prepared dough in plate. Place reserved 3 dough strips along brushed rim, and crimp as desired. In a small bowl, whisk together egg and remaining 1 teaspoon (5 grams) water; lightly brush over dough.

Bake until golden brown, 35 to 45 minutes.