Pudding pie with brownie crust and milk chocolate

Ingredients:

Cake

• 85 g semi-sweet chocolate (not granulated), chopped
• 3 tablespoons unsalted butter
• 1/4 cup light brown sugar
• 1/4 cup granulated sugar
• 1 tsp vanilla extract
• A pinch of fine salt
• 1 large egg
• 0.5 cups premium flour
• 3 tablespoons cocoa powder

Pudding

• 0.5 cup granulated sugar
• 3 tablespoons cocoa powder
• 3 tablespoons cornstarch
• A pinch of fine salt
• 4 large egg yolks
• 2 cups whole milk
• 85g milk chocolate (not granulated), chopped, plus extra for decoration
• 1 tsp vanilla extract

Preparation:

1. Position a rack in the middle of the oven and preheat the oven to 325°F (160°C). Spray the inside of a 9-inch (22 cm) pie pan with cooking spray.
2. Cake:
   
   In a small saucepan over medium heat, stirring constantly, melt the chocolate and butter. Whisk in both sugars, vanilla extract, and salt. Remove from heat and let cool for a few minutes. Add the egg and whisk vigorously until fully incorporated. Add the flour and cocoa powder and knead into a dough; it should be sticky and fairly stiff.
3. Spoon the batter into the center of the pie pan and cover with a piece of plastic wrap large enough to cover the pie pan. Using your fingers, spread the batter over the bottom of the pie pan and up the sides 1/2 inch. Remove the wrap and bake until the top of the cake is crisp and a toothpick inserted into the center comes out clean, about 15 minutes. Check the cake after 10 minutes; if the sides are sagging or the bottom is puffed up, smooth the batter with the back of a large spoon. Transfer the cake to a wire rack to cool for at least 15 minutes.
4. Pudding:
   
   In a medium saucepan, combine the sugar, cocoa powder, cornstarch, and salt. Stir in the egg yolks vigorously, then the milk until smooth. Cook over medium heat, stirring constantly, until the mixture begins to thicken and simmer, about 4 minutes. Add the chopped chocolate and vanilla extract and stir until the chocolate melts, then remove from heat.
5. Pour into a pie pan and cover with plastic wrap, pressing gently onto the surface of the pudding (otherwise a skin will form). Refrigerate until completely set, about 2 hours. Grate some milk chocolate with a vegetable peeler or grater and sprinkle it over the pie filling.