1) Preheat the oven to 350°F. Lightly grease a 9" round pan; or line with parchment, and grease the parchment.
2) Beat the eggs with the sugar, vanilla, baking powder and salt till thick and lightened in color.
3) Combine the butter and milk, and heat till the butter melts and the milk is very hot. Stir the milk to help the butter along. A microwave works well here.
4) Add the flour to the egg mixture alternately with the milk/butter, beating gently just till everything is combined.
5) Pour the thin batter into the prepared pan.
6) Bake the cake for 25 to 30 minutes, until the middle springs back when touched lightly, and a cake tester inserted into the center comes out clean.
7) Remove the cake from the oven, allow it to cool for about 5 minutes, then turn it out of the pan onto a rack.
8) When the cake is completely cool, cut it in half around its circumference, to make a top and bottom half. Put one layer, cut side up, on a serving plate.
9) Make the filling: Place the chocolate chips, corn syrup, vanilla, espresso powder, and cream into a medium-sized microwave-safe bowl, or into a medium saucepan.
10) Heat the mixture till it's very hot; the cream will start to form bubbles. Remove from the heat, and stir until smooth. Beat in the confectioners' sugar.
11) Pour the chocolate filling over the bottom half of the cake. Center the other half of the cake atop the filling.
12) Make the pudding according to the directions on the box, reducing the milk to 1 1/4 cups, and adding 1 teaspoon vanilla.
13) Spread the thickened pudding over the top of the cake. Sprinkle evenly with the coarse sugar.
14) Run a chef's torch back and forth over the surface of the cake to caramelize the sugar. It won't form a solid sheet of caramelized sugar, but will simply become brown and crackly looking.
15) Serve immediately.
16) Looking for a nice final touch atop your cake — without using a chef's torch? Make caramelized sugar, crack it into tiny pieces, and sprinkle on top of the cake. See our tip at right for the details.
Yield: about 8 to 10 servings.