Position rack in center of oven; preheat to 350°F. Generously coat the mini muffin pan with cooking spray.
Place chocolate and butter in a medium microwave-safe bowl. (Alternatively, see “No Microwave?” below.) Microwave on High for 1 minute. Stir well, then continue microwaving on Medium, stirring every 20 seconds, until the remaining chocolate melts completely.
To prepare filling: Stir together granulated sugar, cream and half the espresso mixture in a small microwave-safe bowl. Microwave on High just to steaming, 20 to 40 seconds. (Alternatively, combine in a small saucepan and heat over medium-low heat until just steaming.) Stir until the sugar dissolves. Add corn syrup and half the melted chocolate (reserve the other half for the batter); stir until completely smooth. Cover and transfer to the freezer until cold and firm, about 30 minutes.
To prepare batter: When the filling has been chilling for 20 minutes, whisk egg, oil, vanilla, salt and the remaining espresso mixture in a medium bowl until very smooth. Return the remaining chocolate mixture to the microwave. Microwave on Medium, stirring every 20 seconds, until just warm again (do not overheat). In two batches, whisk the egg mixture into the warm chocolate until well blended. Sift confectioners’ sugar, flour and cocoa over the batter and whisk in just until smoothly incorporated.
To assemble cakes: Remove the filling from the freezer. Spoon half the batter into the mini muffin cups, about 1 rounded teaspoon per cup. Spoon ½ teaspoon filling onto the center of each (reserve the rest for the sauce). Divide the remaining batter evenly among the muffin cups, about 1 rounded teaspoon per cake. Smooth out the batter to cover the filling.
Bake the cakes on the middle rack until the edges look dry and puffed but the centers still look very underdone and puddinglike, 6 to 9 minutes. Let cool on a wire rack until firm, about 2 minutes. Place a cutting board on top of the pan and invert the mini cakes out onto it. If the cakes are stuck to the pan, run a knife around and under them to loosen.
To prepare sauce & serve: Thoroughly stir very hot water into the reserved filling, 1 teaspoon at a time, until very smooth and slightly fluid. Serve the warm cakes drizzled with the sauce.