Preheat oven to 350F and prepare two mini-cupcake trays by lining with cupcake liners.
In small saucepan over medium heat, combine 1 cup milk and dark chocolate, stirring until chocolate is completely melted.
Increase heat to medium-high, stir constantly until mixture begins to bubble and stir in cocoa powder. Remove from heat and whisk until smooth. Set aside.
In KitchenAid, combine canola oil and sugars, beat on medium speed for about 30 seconds, scrape sides of bowl, and stir again on medium-high speed for another 30 seconds -- until ingredients are well-combined (mixture will be very thick and grainy).
Add egg and vanilla, beat well.
In a separate bowl, combine flour, baking powder, baking soda, and salt.
With mixer on low-medium speed, gradually stir in flour mixture to oil/sugar mixture, pausing to scrape down the sides of the bowl occasionally.
Pour the cocoa/dark chocolate liquid into the mixer and beat on medium speed until combined.
Scrape down sides of bowl and add 3 Tbsp milk, stir on medium speed again so all ingredients are combined.
Pour batter into cupcake liners, filling cupcake liners 2/3 of the way full.
Bake on 350F for 15 minutes (toothpick inserted into middle of cupcake should come out clean).
Allow to cool completely before frosting.
Combine brown sugar, butter, and 1/2 cup heavy cream in medium saucepan over medium heat.
Stir until butter is melted and all ingredients are combined.
Increase heat to high and bring liquid to a boil, stirring constantly.
Continue to boil (while still stirring continuously) for two minutes, remove from heat, stirring occasionally so that the mixture does not burn.
Allow candy to cool completely before proceeding.
Once caramel base has cooled, stir in vanilla.
Transfer to KitchenAid (or use electric hand mixer) and gradually stir in powdered sugar, about 1/2 cup at a time, with mixer on low-medium speed.
Pause periodically to scrape down sides of bowl with rubber spatula.
Continue until all sugar has been added, then add 2 Tbsp heavy cream and beat on medium-high until cream has been incorporated.
Pipe onto completely cooled cupcakes and sprinkle with sea salt.