Combine first 3 ingredients in a 4-quart heavy saucepan. Stir in half and-half. Cook, stirring constantly, over medium heat 9 minutes or until mixture comes to a boil and thickens. Boil 2 minutes, stirring constantly. Remove from heat.
Lightly beat egg yolks with a wire whisk. Gradually stir about one-fourth of hot mixture into yolks; add to remaining hot mixture, stirring constantly. Cook, stirring constantly, 3 minutes over medium heat. Remove from heat, and stir in vanilla and butter. Cover and set aside while preparing meringue.
Stir together 3/4 cup brown sugar, 2/3 cup granulated sugar, and water in a 1-quart heavy saucepan. Bring to a boil over medium heat, stirring constantly, until sugar dissolves. Wash down crystals with a small pastry brush dipped in hot water. Bring to a boil over medium heat, and cook, stirring often, until a candy thermometer registers 246° (firm ball stage).
While syrup cooks, beat egg whites and cream of tartar in a large bowl at high speed with a heavy-duty stand mixer until soft peaks form. Pour hot syrup mixture in a heavy stream over beaten egg whites, beating constantly at high speed until stiff peaks form (about 2 to 3 minutes). (Do not overbeat.)
Pour hot filling into pastry shell. Spoon one-third of meringue over hot filling in about 5 large dollops. Using back of a large spoon, spread meringue, sealing to edge of pastry and forming decorative swirls. Spread another third of meringue over pie, using same procedure. Spread remaining third of meringue over pie, and spread to edge.
Bake at 375° for 9 minutes or until meringue is lightly browned. Let cool completely on a wire rack.
Brown Sugar Meringue Pie Pointers:
For best results, time the beating of the egg whites as close as possible to the readiness of the syrup. Since this meringue is baked for only 9 minutes, it's essential that the syrup be 246° when added to the egg whites to bring them to a safe temperature.
Be sure to use a heavy-duty stand mixer with at least a 4-quart bowl to accommodate the large volume of meringue. Don't be tempted to spread on all the meringue at once. It's necessary to apply it in 3 batches in order to fit it all on the pie and ensure maximum volume.