Preheat oven to 350F.
Adjust oven rack to lower-middle position.
Bring a full kettle of water to a boil.
Add parchment paper to a 9-inch cake pan and grease it with butter. You can also cover the insides of a springform pan with foil and grease it with butter. Springform pan works best for this dessert.
In the bowl of an electric mixer fitted with the wire whisk attachment, beat eggs at medium speed for about 5 minutes. They should be light and fluffy.
Add water, coffee, and sugar replacement to a small saucepan over medium heat. Simmer and stir until the sugar replacement is fully dissolved.
Add the chopped chocolate and butter to a large bowl, pour the sugar mixture over and stir until chocolate is fully melted and smooth. Stir in vanilla extract.
Using a rubber spatula, fold ⅓ of the egg foam into the chocolate mixture, stir gently, but quickly, to not cook the eggs, from the warm chocolate mixture. Stir in until only a few streaks of egg are visible; fold in half of remaining foam, then last of remaining foam, until mixture is totally homogenous.
Transfer batter into the prepared pan and smooth surface with a rubber spatula.
Place the cake pan into a roasting pan, transfer to the oven and pour enough boiling water to come about halfway up of the pan.
Bake for about 45 minutes, until the cake has risen slightly, edges are just beginning to set, an instant thermometer inserted in the center of cake must read 140 degrees. The center of the cake should feel settled, when you touch it; the edges should come off slightly from the sides of the pan.
Remove cake pan from the water bath and set on a wire rack; cool to room temperature. Cool to room temperature, remove from springform pan, and cool in the refrigerator for at least 3 hours or overnight before serving.