Prep the pan and oven. Heat the oven to 350°F (175°C).
Line a 9-inch springform pan with parchment on the bottom and lightly grease the sides. Wrap the outside with two layers of foil to prevent leaks during the water bath.
Make the brownie base. Melt butter and semisweet chocolate in a heatproof bowl set over simmering water, stirring until smooth. Off the heat, whisk in both sugars, then the eggs and vanilla.
Sift in flour, cocoa, and salt; fold just until combined. Spread into the pan.
Par-bake the brownie. Bake 12–15 minutes until set around the edges but slightly soft in the center. Cool 10 minutes while you make the filling.
Reduce oven temp to 325°F (160°C).
Mix the cheesecake filling. In a large bowl, beat cream cheese and sugar on medium speed until smooth and fluffy, about 2 minutes. Mix in sour cream, vanilla, and peppermint extract. Add flour and salt; beat on low just to combine.
Beat in eggs one at a time on low until fully incorporated. Do not overmix.
Assemble and prepare the water bath. Pour the cheesecake batter over the warm brownie base. Tap the pan gently to release air bubbles.
Place the springform pan in a larger roasting pan. Add hot water to the roasting pan until it reaches halfway up the sides of the springform.
Bake the cheesecake. Bake at 325°F for 55–70 minutes, until the edges are set and the center still has a slight wobble. The top should look matte, not wet.
Turn off the oven, crack the door, and let it rest inside for 45 minutes.
Cool and chill. Remove from the water bath. Run a thin knife around the edge to loosen. Cool to room temperature, then cover and chill at least 6 hours or overnight.
Make the ganache. Heat the cream until steaming (not boiling).
Pour over chopped chocolate and let sit 2 minutes. Stir until smooth; add butter if using. Let thicken slightly until it coats a spoon.
Top and garnish. Release the cheesecake from the pan and set on a serving plate.
Spread ganache over the top, letting it gently drip over the edges. Sprinkle with crushed peppermint and any optional garnishes. Chill 15–20 minutes to set before slicing.
Serve. For clean slices, warm a sharp knife under hot water and wipe between cuts.
Store leftovers covered in the fridge.