Heat the oven to 375°F with a rack in the middle position. In a medium bowl, whisk together the flour and salt, then make a well in the center; set aside.

In a small saucepan, combine the butter, shortening and ¼ cup water. Bring to a simmer over medium-high, stirring to melt the solids. As soon as they’re melted and the mixture is simmering, pour it into the well of the dry ingredients. Working quickly, stir with a silicone spatula until the dry ingredients are evenly moistened and without any dry patches; the dough will be very soft and resemble wet mashed potatoes. Turn it out onto a large sheet of plastic wrap and, using your hands, form it into a 6- to 8-inch disk.

Cover the dough disk with another large sheet of plastic wrap and roll it into a 12-inch round of even thickness. Peel off the top sheet of plastic. Using the bottom sheet, carefully flip the round into a 9-inch pie plate, centering it as best you can. Ease the dough, still on the plastic, into the corners and up the sides of the pie plate. Carefully peel off the plastic. If needed, patch any tears in the dough. Trim the excess dough and flute or crimp the edge. Carefully line the dough with a large sheet of foil, gently pressing it into the corners and up the sides, then fill with about 2 cups pie weights.

Bake until the dough is set, about 20 minutes. Carefully lift out the foil with weights, then prick the pie shell all over with a fork to deflate any air bubbles and prevent additional ones from forming. Bake until the shell is lightly browned, another 12 to 15 minutes. Transfer to a wire rack and let cool. Reduce oven to 350°F.

In a blender, combine 135 grams (1½ cups) of the shredded coconut, the coconut milk, sugar, cornstarch, lime juice, vanilla and salt. Puree, scraping the blender jar once or twice, until the mixture is as smooth as possible, 1 to 2 minutes. Add the eggs and blend until combined, about 10 seconds.

Pour the puree into a large bowl. Stir in the remaining 90 grams (1 cup) shredded coconut, then pour the mixture into the pie crust. Bake until the filling is lightly browned and puffed at the edges and the center is slightly domed, 55 to 60 minutes.

Cool on a wire rack; the filling will settle during cooling. Serve slightly warm or at room temperature.