Preheat the oven to 350°F. Spray a 9 inch springform pan with nonstick spray, line the bottom with a round of parchment, and wrap the outside of the pan in a double layer of aluminum foil to catch any drips.
In a medium bowl whisk together the melted butter and 1 cup granulated sugar until glossy and well combined.
Whisk in 2 eggs, one at a time, then add 1 teaspoon vanilla bean paste and mix until the mixture looks thick and shiny.
Sift in the cocoa powder, flour, baking powder, and 1/8 teaspoon salt, then whisk just until a thick, smooth brownie batter forms.
Scrape the brownie batter into the prepared springform pan and smooth it into an even layer, spreading all the way to the edges.
Bake the brownie base for 18 to 20 minutes, until the edges are set and the center looks slightly puffed and shiny but still soft.
Remove the pan from the oven and let the brownie cool in the pan on a wire rack for 10 to 15 minutes while you prepare the cheesecake filling. Reduce the oven temperature to 325°F.
In a large bowl beat the softened cream cheese with a hand mixer or stand mixer on medium speed until completely smooth and creamy, about 2 minutes, scraping the bowl as needed.
Add 3/4 cup granulated sugar and beat until the mixture looks lightened and fluffy, about 1 to 2 minutes.
Beat in 3 eggs, one at a time, on low speed, mixing just until each egg is incorporated before adding the next.
Add the sour cream, 1/3 cup heavy cream, remaining 1 teaspoon vanilla bean paste, and remaining 1/8 teaspoon salt, then mix on low speed until the batter is smooth and pourable with no visible lumps.
Tap the bowl gently on the counter a few times to release excess air bubbles, then pour the cheesecake filling over the slightly cooled brownie layer in the springform pan.
Smooth the top of the cheesecake with an offset spatula and tap the pan firmly on the counter 3 to 4 times to bring more air bubbles to the surface.
Place the springform pan on a rimmed baking sheet and bake at 325°F for 45 to 50 minutes, until the outer 2 inches look set and matte and the center jiggles in a wide, soft wobble when you gently shake the pan.
Turn off the oven, crack the oven door several inches, and let the cheesecake cool inside the oven for 30 minutes to help prevent cracks.
Remove the pan from the oven and transfer it to a wire rack; cool the cheesecake to room temperature, about 1 1/2 to 2 hours.
Once the cheesecake reaches room temperature, cover the pan loosely with plastic wrap and refrigerate for at least 4 hours, or until thoroughly chilled and set.
For the chocolate ganache, place the chocolate chips in a heatproof bowl. Pour the 1/2 cup hot heavy cream over the chocolate, let sit for 2 minutes, then whisk until smooth and glossy.
Let the ganache cool until thickened but still pourable, about 10 to 15 minutes at room temperature, stirring occasionally.
Remove the chilled cheesecake from the refrigerator, run a thin knife around the edge, then release and remove the sides of the springform pan.
Pour the cooled ganache over the top of the cheesecake, spreading gently with an offset spatula and allowing a few drips to run over the edges for a bakery style look.
Immediately scatter rainbow sprinkles over the ganache for a birthday cake finish, then return the cake to the refrigerator for 20 to 30 minutes to let the ganache firm slightly.
When ready to serve, slice the Brownie Cheesecake Birthday Cake with a large, sharp knife dipped in hot water and wiped dry between cuts for the cleanest slices.
Store any leftover cake covered in the refrigerator for up to 4 days.