With a fork, prick bottom and sides of unbaked pastry in pan at about 1-inch intervals. Bake in a 375° regular or convection oven until golden, 15 to 20 minutes. Let cool on a rack.
Scatter coconut over bottom of pastry.
In a 2- to 3-quart pan, mix 1 cup sugar, 1/4 cup cornstarch, milk, and egg yolks until well blended. Stir over medium-high heat until mixture boils and thickens, 10 to 15 minutes. Remove from heat and stir in vanilla. Pour over coconut in crust.
In a small bowl, mix remaining 1/3 cup sugar and 1 1/2 teaspoons cornstarch.
In a deep bowl, with a mixer on high speed (use whisk attachment if available), beat egg whites, cream of tartar, and salt until very foamy. Add sugar-starch mixture, 1 tablespoon every 30 seconds, and continue to beat until meringue holds stiff, glossy peaks.
Spoon meringue onto hot pie filling. With a spatula, spread meringue evenly to rim of pie shell.
Bake in a 325° regular or convection oven until meringue is lightly browned, 15 to 20 minutes.
Let pie cool on a rack about 3 hours. Cut into wedges and serve, or invert a large bowl over pie (it shouldn't touch meringue) and chill up to 1 day.