Make the Cake: Preheat oven to 350 degrees F. Grease a 9x13-inch pan; set aside.
In a medium bowl, sift together the flour and baking soda, then stir in the salt.
Using an electric mixer, beat the butter and sugar on medium speed for 5 minutes, until light and fluffy. Scrape down the bowl, then add the vanilla and beat in the eggs, one at a time, until incorporated and the batter is smooth.
In a medium bowl or large 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, beginning and ending with the flour. With a rubber spatula, scrape down the bowl and ensure that the batter is well-mixed.
Pour the batter into the prepared pan, smooth the top, and bake for 20 to 30 minutes, until a cake tester inserted in the center comes out clean. Cool completely in the pan on a wire rack before frosting.
Make the Frosting: Place the chocolate, butter and espresso powder in a large bowl. Pour the cream into a small saucepan and place over medium heat until it simmers. Remove from the heat and pour it evenly over the chocolate mixture. Let stand for 1 minute, then gently stir with a rubber spatula until everything is melted and smooth. Stir in the Kahlua and vanilla extract until thoroughly combined. Refrigerate for 30 minutes.
Scrape down the bowl and beat the frosting with a handheld mixer on high speed for ONLY 15 to 20 seconds, until the mixture forms soft peaks (if you overbeat it, it will curdle). Immediately spread on the cake and serve. The cake can be stored, covered, at room temperature for up to 4 days.