1. Preheat oven to 375 degrees. Make the profiteroles: Sift together flour, cocoa, and 1/8 teaspoon salt. Bring milk, butter, and granulated sugar to a boil in a medium saucepan, stirring constantly until sugar dissolves. Remove from heat, and immediately stir in flour mixture using a wooden spoon. Return saucepan to medium heat; cook, stirring constantly, until dough forms and flour lumps dissolve, about 2 1/2 minutes.

2. Transfer dough to the bowl of a mixer, and beat on medium speed until slightly cooled, about 2 minutes. Beat in 2 eggs, 1 at a time. Test dough by touching it with your finger and lifting to form a string. If string does not form, the batter needs more egg. Lightly beat remaining egg, and add to dough, 1 teaspoon at a time, testing after each addition. Beat until batter is smooth and glossy, about 2 minutes more.

3. Using a pastry bag fitted with a 1/2-inch round tip, pipe about twenty 1 1/2-inch rounds of batter onto a parchment-lined baking sheet, spacing about 1 inch apart. Bake until puffed and cooked through, about 20 minutes. Transfer to a wire rack; let cool completely.

4. Meanwhile, make the peppermint cream: Bring milk, 1/4 cup granulated sugar, and 1/4 teaspoon salt to a boil in a medium saucepan. While milk is heating, whisk together yolks and remaining 1/4 cup granulated sugar in a medium bowl. Whisk in cornstarch, 1 tablespoon at a time, until combined. Once milk starts to boil, slowly pour it into yolk mixture, whisking constantly. Return entire mixture to saucepan; whisk. Cook over medium-high heat until thick, about 1 1/2 minutes. Immediately remove from heat, and strain through a fine sieve into a bowl. Stir in peppermint. Press plastic wrap directly onto surface of cream, and refrigerate until cool, about 1 hour.

5. Whisk peppermint cream until smooth. Whisk heavy cream in a medium bowl until medium-stiff peaks form. Working in 2 batches, fold whipped cream into peppermint cream.

6. Make the chocolate sauce: Heat heavy cream and butter in a small saucepan over medium-high heat until butter melts. Add sugars, whisking until dissolved. Whisk in cocoa and a pinch of salt. Strain chocolate sauce through a fine sieve into a bowl. Let stand until ready to use.

7. Assemble the profiteroles: Slice profiteroles in half horizontally. Sandwich about 1 tablespoon peppermint cream between each. Pour warm chocolate sauce over profiteroles just before serving.