To make the fudge sauce, combine the butter, cream, brown sugar, corn syrup, cocoa, and salt in a small saucepan over medium heat and bring to a simmer. Simmer on low, stirring, for 3 to 5 more minutes after everything has melted, then remove from heat and stir in the chocolate. Stir in vanilla.
For the crust, heat oven to 350and lightly spray the bottom of a 9-inch springform pan with cooking spray. Wrap the bottom in foil (in case some of the brownie batter seeps out.
Pour roughly ⅓ of the batter into the pan and spread out evenly. You don't want a huge crust, this recipe is about the cheesecake more than the brownie. Bake the crust for about 15-20 minutes. You want it to be basically just baked, but not raw in the center. Remove from oven and cool completely.
Turn the oven down to 325.
Boil a pot of water. Place a large metal baking dish or a catering half pan, on the bottom rack in the oven, and make sure you have a rack for the cheesecake in the middle of the oven. Take your boiling water and CAREFULLY pour it into the dish on the bottom rack of the oven and close the oven right away.
To make the cheesecake, melt the chocolate in a bowl over a saucepan of boiling water, stirring occasionally until all the chocolate is melted (basically a double boiler).
In the bowl of a stand mixer (or a large bowl if you are using a hand mixer), beat with the whisk attachment the cream cheese for a good three minutes until it is light and fluffy. Add the sugar and beat for another two minutes. Beat in the cocoa powder (do this on low so it doesn't blow all over) and the peppermint extract. Add the eggs, one at a time, and beat for a minute after each egg. With the mixer going, add the melted chocolate and beat until incorporated and smooth.
Pour the filling into springform, it will be really high.
Bake on middle rack for 50 minutes, keeping the oven closed as much as possible (resist the urge to peek). Turn off the oven, and crack it open. Let cheesecake cool down in oven for at least an hour, or up to three hours. Finish cooling at room temperature, and then refrigerate until serving. Drizzle with your the hot fudge sauce when cold and serve extra at the table.