In a heavy-bottomed saucepan, heat 1/2 cup sugar over medium heat until it begins to melt around the edges. Swirl the pan occasionally (don’t stir) to help the sugar melt evenly. When it turns a deep amber color, quickly pour the caramel into pudding molds or ramekins. Swirl each mold so the caramel coats the bottom evenly. Let it sit to harden while you prepare the custard.

In a saucepan, heat milk over medium heat until warm (do not boil). Add the chopped chocolate to the warm milk and stir until completely melted and smooth. Remove from heat. In a bowl, whisk together eggs, egg yolks, and sugar until smooth and pale. Slowly pour the warm chocolate milk mixture into the egg mixture, whisking constantly to prevent curdling. Stir in vanilla extract. Strain the custard mixture through a fine sieve to remove any lumps or bubbles.

Pour the chocolate custard mixture over the hardened caramel in each ramekin. Place the ramekins in a deep baking dish and pour hot water into the dish until it reaches halfway up the sides (a water bath). Bake in a preheated oven at 325°F (160°C) for 40–45 minutes, or until the center is just set but still slightly jiggly.

Remove from the oven and let the puddings cool at room temperature. Refrigerate for at least 4 hours, or overnight, until fully chilled. To serve, run a thin knife around the edge of each pudding, invert onto a plate, and let the caramel sauce drizzle over the top.