Preheat the oven to 350 degrees. Make the cakes: Brush the four 1 1/4 cup brioche molds or jumbo muffin cups with the 2 tablespoons of melted butter. Dust the molds with cocoa powder and tap out the excess.
Whisk the flour, baking soda, baking powder, and salt in a small bowl. Bring milk and 3/4 cup water to a simmer in a saucepan over medium heat; set aside.
Combine the vegetable oil, 4 tablesoons room-temperature butter and sugar in a stand mixer and beat with the paddle attachment on medium-high speed until fluffy, about 4 minutes, scraping down the bowl and beater as needed. Add 1/2 cup cocoa powder and the vanilla; beat 1 minute on medium-low speed, then add the remaining egg and beat 1 more minute.
With the mixer on low speed, gradually beat in the flour mixture, then the hot milk mixture. Finish mixing the batter with a rubber spatula until combined. Divide the batter evenly among the molds, filling each slightly more then three-quarters of the way.
Transfer the mold to a baking sheet and bake until tops of the cakes are domed and the centers are barely set, 25 to 30 minutes. Transfer the baking sheet to a rack; let the cakes cool until they pull away from the molds, about 30 minutes.
MAKING THE FILLING:
Microwave the chocolate, cream, butter, and corn syrup in a microwave safe bowl in 30 second intervels, stirring each time, until chocolate starts to melt, 1 minute 30 seconds. Let sit 3 minutes, then whisk until smooth. Reheat before using, if necessary.
Use the tip of the paring knife to gently loosen the cakes from the molds, then invert the cakes onto a cutting board.
Use the knife to cut a 1 1/2 inch circle on the top of each cake, cutting almost to the bottom.
Hollow out the cakes with a spoon; save the scraps. Wrap the cakes with plastic wrap and microwave until steaming, 1 minute.
Drizzle plates with caramel, then unwrap the cakes and place on top. Pour about 3 tablespoons filling into each cake.
Plug the hole with the cake scraps. Save or discard any remaining scraps.
Top each cake with a scoop of ice cream. Spoon more chocolate sauce on top, spreading it thin so it hardens into a shell.