In a mixing bowl, combine 1 cup of all-purpose flour, ⅓ cup of icing sugar, ¼ cup of cocoa powder, and ¼ teaspoon of salt. Mix these dry ingredients until well blended. Then, add 1 stick of melted butter and ½ teaspoon of vanilla extract; use your hands to blend until the mixture becomes crumbly. Press this dough into a tart pan, ensuring an even layer, and pre-bake in a preheated oven at 350°F (175°C) for about 10 minutes, or until slightly firm.

While the tart base is baking, whisk together 3 egg yolks and 2 whole eggs in a mixing bowl. Gradually add 1 cup of white sugar, ⅓ cup of cornstarch, and ½ teaspoon of salt, mixing until smooth. Slowly stir in 2 ½ cups of whole milk and 1 ¾ cups of heavy cream, ensuring everything is well combined. Afterwards, melt 7 ounces of dark chocolate, then gently fold it into the custard mixture until fully incorporated and silky.

Once the tart base is ready, remove it from the oven and pour the chocolate custard mixture into it. Set the tart pan into a larger baking dish and fill the larger dish with hot water, creating a water bath. Bake the French chocolate flan at 325°F (163°C) for about 50-60 minutes, or until the custard is just set but still slightly jiggly in the center. Check for doneness by gently shaking the pan; the edges should be firm while the middle remains soft.

After baking, carefully remove the tart pan from the water bath and let the flan cool at room temperature for about 30 minutes. Once cooled, refrigerate for at least 2 hours, allowing the flavors to meld and the texture to set. Before serving, give the top a glossy finish by brushing it with 1 tablespoon of jam, and enjoy a slice of your beautifully made French chocolate flan with friends and family!