1 1/2 c. all-purpose flour (spooned and leveled)
1/2 c. (1 stick) cold unsalted butter, cut into small pieces
1/3 c. sugar
1/4 tsp. salt
For the Filling:
1 1/2 c. sweetened shredded coconut (4 oz.)
2 1/2 c. milk
1/2 c. sugar
3 T. cornstarch
Pinch of salt
3 large egg yolks
1 tsp. vanilla extract
1 c. heavy cream
Make crust: Preheat oven to 350. In a food processor, pulse flour, butter, sugar, and salt until moist crumbs form; transfer to a 9-inch tart pan with removable bottom. With floured fingers, press evenly into bottom and up sides of pan. Freeze until firms, 15 minutes. Prick crust all over with a fork. Bake until golden, 25 minutes, pressing down with a spoon if it puffs up; cool completely. Leave oven on. Make filling: spread 3/4 cup coconut on a rimmed baking sheet. Bake until golden, tossing occasionally, 5 to 7 minutes. Set toasted coconut aside.
In a medium saucepan, heat 2 cups of the milk and remaining 3/4 cup coconut until milk just beings to simmer. Remove form heat, cover, let stand 10 minutes. Strain through a sieve, and return milk to saucepan, discard coconut.
In a medium bowl, whisk together sugar, cornstarch, salt, egg yolks, and remaining 1/2 cup milk until blended. Gradually whisk into saucepan.
Bring to a simmer over medium-low heat, whisking constantly until mixture begins to thicken, about 3 minutes. Cook, whisking, 1 minute. Transfer to a bowl, add vanilla. Cover and refrigerate until cool, 45 minutes.
In a medium bowl, with an electric mixer, beat cream on medium until soft peaks form.