Preheat the oven to 350°F.
For the filling: In the bowl of a stand mixer fitted with the paddle attachment, beat the cream cheese, granulated sugar, and buttermilk on medium speed until smooth, 1 to 2 minutes. Scrape down the sides and bottom of the bowl and beat in the egg and vanilla until well combined, about 30 seconds. (You should have a heaping 2 cups of filling.)
For the cake: In a medium bowl, whisk together the flour, cocoa powder, baking soda, and salt. In a large bowl, whisk together the granulated sugar, vegetable oil, buttermilk, vanilla, vinegar, and eggs until well combined. Whisk in the food coloring. Gradually add the flour mixture to the buttermilk mixture, whisking until just combined.
Spray a 10-cup Bundt pan with baking spray with flour. Add about 3 cups of the batter to the bottom of the Bundt pan, spreading until even. Spoon the filling over the batter, leaving a ½-inch border along the outer edge and the center of the pan. Top evenly with the remaining batter and smooth with a rubber spatula. Tap the pan on the counter to release any air bubbles.
Bake the cake until a wooden pick inserted in the center comes out clean, 1 hour to 1 hour 10 minutes. Transfer to a cooling rack and allow to cool in the pan for 30 minutes before inverting onto a serving platter. Allow to cool completely, at least 2 hours.
For the glaze: In a medium bowl, whisk together the cream cheese, powdered sugar, buttermilk, butter, and vanilla until smooth. Add additional buttermilk, 1 teaspoon at a time, until the glaze is thick but pourable.
Spoon or pour the glaze on top of the cake, allowing it to drip down the sides. Decorate with sprinkles, if you like. Allow the cake to set for about 20 minutes before slicing and serving.