Place milk and sugar in a heavy saucepan and bring just to a boil, stirring until sugar is dissolved.

In a medium bowl, whisk together the eggs and cornstarch until smooth. Whisk about a cup of the hot milk mixture into the eggs to temper, then add the egg-milk mixture back to the pot. Reduce the temperature to medium-low and continue to cook, whisking constantly, until mixture is very thick and any starchy taste is gone, 5 to 10 minutes.

Remove from heat and whisk in the butter and coconut extract, then immediately strain the custard through a very fine strainer into a bowl. Set the bowl inside a larger bowl that is filled with an ice bath and stir occasionally until the custard has cooled slightly. Transfer the custard to the cooled pie shell and refrigerate, covered with plastic wrap, until thoroughly chilled. When the custard is firm and pie is completely cooled, top the pie with the Lime Meringue or, if preferred, with whipped cream.

Cashew Pie Crust

Cut the butter into small pieces and place in the freezer until very well-chilled.

Place flour, cashews, salt, and lime zest in a food processor. Add the chilled butter and pulse the processor until the butter is the size of small peas. Add the egg yolk and vanilla extract and pulse until dough just begins to come together. You may need to add a bit of ice water, a teaspoon at a time, depending on the size of your egg yolk. Gather the dough into a flattened disc, wrap in plastic wrap, and refrigerate for at least 1 hour and up to overnight.

On a lightly floured surface and using a floured rolling pin, roll the dough out to a thickness of about 1/8-inch and carefully place in pie dish. Trim the excess dough to about ½-inch and tuck edges into sides of pie dish. Crimp the edges decoratively and refrigerate the pie crust for 1 hour.

Preheat the oven to 400°F and dock the pie crust all over using the tines of a fork. Line the pie crust with parchment paper and fill with pie weights. Bake the pie shell in oven until the sides are set, 8 to 12 minutes, then remove the parchment and pie weights and continue to cook until the crust is golden brown and flakey, 8 to 10 minutes longer. Set crust aside to cool on a wire rack.

Lime Meringue

For the syrup, place the sugar, water, and lime juice in a heavy bottomed sauce pan and cook over high heat until the mixture reaches between 230°F and 240°F on a candy thermometer, about 4 to 5 minutes. Remove from the heat.

While the sugar-lime mixture is cooking, place the egg whites and the 1 tablespoon plus 1 teaspoon sugar in the bowl of an electric mixer fitted with the whisk attachment and beat on high speed until the egg whites hold soft peaks.

Slowly drizzle the hot syrup into the egg whites while continuing to beat on high speed, until the meringue cools to lukewarm, about 5 minutes. Fold in the lime zest. Spread the meringue over the top of the cooled coconut pie.