Preheat the oven to 300°F (150°C). Grease a 9-inch round cake pan. Line the bottom with parchment paper (with extended edges for easy lifting) and lightly grease again.
In a small saucepan over medium heat, combine water, salt, and sugar. Whisk occasionally until fully dissolved, then remove from heat and set aside.
Melt the chocolate chips using a microwave or double boiler. Transfer the melted chocolate to a stand mixer bowl.
Cut the butter into tablespoon-sized pieces. Beat into the melted chocolate one piece at a time until fully incorporated.
Pour the warm sugar-water mixture into the chocolate mixture and mix until smooth.
Beat in the eggs one at a time, ensuring each egg is fully incorporated before adding the next. Continue until the batter is smooth.
Stir in the vanilla extract and mix just until combined.
Pour the batter into the prepared cake pan.
Place the cake pan inside a larger baking pan and transfer to the oven. Carefully pour boiling water into the outer pan until it reaches halfway up the sides of the cake pan.
Bake for about 45 minutes. The edges should be set while the center remains slightly soft with a gentle jiggle.
Remove the cake from the water bath and let it cool completely on a wire rack.
Refrigerate the cake in the pan for at least 6 hours or overnight for best texture.
Lift the cake out using the parchment paper and transfer to a serving plate. Dust with powdered sugar and serve with berries or ice cream if desired.