Preheat the oven to 400 degrees F. In a saucepan, whisk the 2 cups of the milk and 3/4 cup sugar together. Place the pan over medium heat and bring the liquid up to a simmer. Whisk the egg yolks together. Temper the hot milk into the egg yolks. Whisk the egg mixture into the hot milk mixture. In a small bowl, dissolve the cornstarch in the remaining milk, making a slurry. Whisk the slurry into the hot milk mixture. Bring the liquid up to a boil and reduce to a simmer. Cook the mixture, stirring constantly, until the filling is thick, about 4 to 6 minutes. Fold in the vanilla, coconut, and butter. Mix well. Pour the filling into the prepared pan and cool the pie completely. Using an electric mixer with a whip attachment, whip the egg white to soft peaks. Add the remaining sugar and whip the egg white to stiff peaks. Spread the egg whites over the top of the pie. Place the pie in the oven for about 3 to 4 minutes, or until the meringue is golden brown.