Meringue Pie Shell: Heat oven to 225°F. Beat egg whites and cream of tartar in mixer bowl with whisk attachment on high speed until foamy. Beating constantly, add sugar, 2 Tbsp at a time, beating after each addition until sugar is dissolved before adding the next. (Rub a bit of meringue between thumb and forefinger; it should feel completely smooth.) Continue beating until whites are glossy and stand in stiff peaks. Beat in vanilla.
Spread meringue on bottom and sides of greased 9-inch pie plate with back of spoon, building up edge to form a rim.
Bake in 225°F oven until firm and wooden pick inserted in center comes out clean, 1 to 1 1/2 hours. Turn off oven. Let dry in oven, with door closed, until cool and crisp, at least 1 hour. Store, in pie plate, in tightly sealed container at room temperature. Fill just before serving.
Key Lime Filling: Beat egg yolks, milk, lime juice and lime peel in a heavy saucepan until blended. Cook over low heat, stirring constantly but gently, until mixture is just thick enough to coat a metal spoon with a thin film and temperature reaches 160°F, about 15 minutes. Do not allow to boil. Remove from heat immediately.
Cool quickly. Set pan in larger pan of ice water; stir occasionally and gently for a few minutes to hasten cooling. Press piece of plastic wrap onto surface of custard to prevent a "skin" from forming. Refrigerate until thoroughly chilled, at least 1 hour.
Spread lime custard evenly in meringue pie shell before serving.