Preheat oven to 325 degrees F (165 degrees C). Grease a 10-inch cake pan with oil.

Combine chocolate, butter, and sugar in a double boiler over simmering water; cook, stirring frequently and scraping down the sides with a rubber spatula, until melted and smooth, about 5 minutes.

Pour chocolate mixture into the bowl of a stand mixer fitted with the paddle attachment; beat on low speed until cooled, about 5 minutes. Increase speed to medium; beat in eggs one at a time. Add water and vanilla extract; beat on low speed until blended. Pour into the prepared pan.

Place cake pan inside a large baking dish. Pour enough water into the baking dish to come halfway up the sides of the cake pan.

Bake cake in the preheated oven until a knife inserted into the center comes out clean, about 1 hour. Allow cake to cool in the water bath, about 1 hour. Transfer to the refrigerator; chill until firm, 8 hours to overnight.