In a large microwave-safe bowl, combine unsweetened chocolate and 1/2 cup half-and-half. On the defrost or low heat setting, microwave 1 minute. Whisk lightly and then repeat until mixture is melted and smooth. Make sure mixture is warmed just enough to melt the chocolate; the hotter this chocolate cream gets, the less creamy the baked pie will be.
Place brown sugar in a separate bowl and break up any lumps. Whisk in eggs, one at a time, mixing until smooth. Whisk in corn syrup and vanilla. Gradually whisk about one-quarter of this egg mixture into chocolate and blend well. It is important to whisk in the eggs slowly and steadily. Continue adding eggs while whisking until all are combined. Pour chocolate custard into pie shell.
Bake in a preheated 350-degree oven about 45 minutes, until the filling is puffy and set around the edges but the center still shimmies. If you want a denser pie, bake 5 to 10 minutes longer, until it puffs up evenly. Transfer to a wire rack and let cool slightly, then cover and refrigerate until set completely, 3 to 4 hours.
Make Sweetened Whipped Cream: In a chilled mixing bowl with chilled beaters, whip 1 1/3 cups heavy whipping cream with an electric mixer on medium speed until soft peaks form. Gradually beat in 2 tablespoons sugar and 1/4 teaspoon vanilla and continue beating until stiff peaks form. Makes 2 3/4 cups.
Make Chocolate Sauce: Place chocolate chips and remaining 1 tablespoon half-and-half in a microwave-safe dish and microwave on low heat until melted, 30 to 60 seconds. Stir until smooth.
Mound Sweetened Whipped Cream over pie, spreading to edges. Drizzle warm chocolate sauce over whipped cream in a random pattern. Serve at once or refrigerate up to a day. Makes 1 (9-inch) pie, 8 servings.