Separate the egg yolks from the whites. Place the yolks in a small bowl and beat with a fork or whisk until mixed. In a small bowl, place the yolks. Beat the egg yolks with a fork or wire whisk until mixed. Stir in 1-1/2 cups milk; set aside. (Save the whites for another recipe.)
In a 2-quart saucepan, stir 1/3 cup granulated sugar, the cornstarch and 1/8 teaspoon salt until mixed. Gradually stir egg mixture into sugar mixture. Cook over medium heat, stirring constantly, until mixture thickens and boils. Boil and stir 1 minute; remove from heat. Stir in 2 teaspoons vanilla. Press plastic wrap on surface of filling to prevent a tough layer from forming on top. Refrigerate at least 2 hours until set but no longer than 24 hours. While filling is chilling, continue with recipe.
Preheat the oven to 350°. Grease and flour the bottom of a 9-inch round cake pan or line bottom with parchment paper.
In a large bowl, beat all cake ingredients with an electric mixer on low speed 30 seconds, stopping frequently to scrape batter from side and bottom of bowl with a rubber spatula. Beat on high speed 3 minutes, stopping occasionally to scrape bowl. Pour batter into the pan; use a rubber spatula to scrape batter from bowl, spread batter evenly in pan and smooth top of batter.
Bake about 35 minutes or until a toothpick inserted in the center comes out clean. Cool cake in pan on a cooling rack 20 minutes, then remove onto cooling rack to finish cooling completely, about 1 hour.
In a 1-quart saucepan, melt 3 tablespoons butter and the chocolate over low heat, stirring occasionally. Meanwhile, in 1-cup glass measuring cup, microwave the water uncovered on High 15 to 30 seconds or until hot. Remove chocolate mixture from heat. Stir in the powdered sugar and 3/4 teaspoon vanilla. Stir in 3 tablespoons hot water. Stir in additional water, 1 teaspoon at a time, until icing is smooth and thin enough to spread.
To split cake horizontally in half, mark middle points around side of cake with toothpicks. Using toothpicks as a guide, cut through the cake with a long, sharp knife, using a back-and-forth motion. On a serving plate, place bottom layer with the cut side up. Spread filling over bottom layer. Top with top of cake, cut side down.
Spread glaze over top of cake, using a metal spatula or back of a spoon, letting some glaze drizzle down side of cake. Refrigerate uncovered until serving. Store any remaining cake covered in the refrigerator.