Preheat the oven to 350 degrees F. Grease a 9-by-13-by-2-inch baking pan. Line with parchment paper, then grease and flour the pan.
Place the butter and sugar in the bowl of an electric mixer fitted with the paddle attachment and beat on medium speed for 4 to 5 minutes, until light and fluffy, scraping down the bowl. On medium speed, add the vanilla, then beat in the eggs, one at a time, until incorporated and the batter is smooth.
Sift the flour, baking soda, and salt together in a medium bowl. In another bowl or a liquid measuring cup, whisk together the hot water, cocoa powder, and espresso powder until smooth. Add the half-and-half and whisk until smooth. With the mixer on low, add the flour and chocolate mixtures alternately in thirds, starting and ending with the flour. With a rubber spatula, scrape down the bowl to be sure the batter is well mixed.
Pour the batter into the prepared pan, smooth the top, and bake for 25 to 35 minutes, until a cake tester inserted in the center comes out clean. Cool completely in the pan. Turn out onto a flat platter or board and frost the top with the Mocha Frosting. Cut in squares and serve.
Place the chocolate, butter, and espresso powder in a bowl. Heat the cream to simmer and pour it over the chocolate mixture, stirring occasionally, until the chocolate is melted. (If the chocolate isn't melted, microwave the mixture for 15 seconds.) Stir in the Kahlua and vanilla and stir until the mixture is smooth. Cover and refrigerate for 30 minutes only, until cool but not cold.
Scrape down the bowl and beat the frosting with a handheld mixer on high speed for 15 to 20 seconds only, until the mixture forms soft peaks. (If you overbeat it, it will curdle!) Spread on the cake immediately with a metal spatula.