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Preheat the oven to 350°F. Lightly grease and flour (or grease, then line with parchment, then grease again) two 8" x 2" round cake pans. Note: These pans need to be at least 2" tall; if you have non-standard, shorter 8" pans, substitute 9" round pans.
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To make the cake: Whisk together the dry ingredients. Add the eggs, oil, and vanilla, beating until smooth.
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Gradually add the water, beating until smooth.
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Pour the batter into the prepared pans.
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Bake the cakes for 35 to 38 minutes (a bit less if you use a 9" pan), or until a cake tester inserted into the center comes out clean. Cool the cakes in the pan for 15 minutes, then turn them out of the pans to cool completely on a rack.
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To make the filling: Mix the sugar, peanut butter, and vanilla until crumbly. Add the milk and mix until smooth, adding more milk if necessary to make a spreadable filling.
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When ready to serve the cake, divide the cooled layers in half horizontally, to make four layers. Place one layer on a serving plate, and spread with filling. Repeat with the next two layers. Top with the final cake layer.
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To make the icing: Combine the chocolate chips, cream, and corn syrup in a microwave-safe measuring cup. Microwave until the cream is very hot, and the chips are soft.
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Stir until the chips melt, and the mixture is smooth.
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Spread the icing atop the cake, letting it drizzle down the sides.
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Serve in thin slices; the cake is very rich.
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Refrigerate any leftovers for several days or freeze for longer storage.
