Preheat oven to 325°F (165°C).
Line the bottom of an 8- or 9-inch spring-form pan with parchment.
Grease the inside walls and the parchment with butter. Take a very large piece of aluminum foil and tightly wrap the entire outside of the pan (this is to prevent water from seeping into the pan once placed in the water bath).
Using either a hand mixer or a KitchenAid stand mixer, beat the eggs until almost doubled in size, about 5–7 minutes.
Meanwhile, using either a double boiler or the microwave, melt the chocolate, butter, and espresso together (if using the microwave, use a glass bowl, and heat in 30-second intervals, removing and stirring after each interval).
Bake for 18–22 minutes, or until an instant read thermometer registers 140°F (60°C). Trust me, you don’t want to overcook this cake!
Remove pan from water bath, and let sit on a rack to cool.
Once it’s reached room temperature, remove the sides of the spring form pan, and invert cake onto a plate so as to remove the paper/spring-form bottom.
Turn right side up onto a cake stand or platter.