Preheat oven to 325°F (170°C). Line bottom and sides of a 9-inch springform pan with parchment paper.

For cake base: In the bowl of a stand mixer fitted with the paddle attachment, beat granulated sugar, brown sugar, oil, and eggs at medium speed for 2 to 3 minutes. Add sour cream and vanilla, and beat until smooth. Add flour, cocoa, baking powder, baking soda, and kosher salt. Fold until combined.

In a small microwave-safe bowl, heat milk on high until bubbles form around the sides of the bowl, about 45 seconds. Add to batter, and stir until fully combined.

For cheesecake: Beat cream cheese in a large mixing bowl with an electric mixer on medium speed for 1 minute. Then add sugar, eggs, vanilla extract, and sour cream. Mix until combined and smooth, 1 to 2 minutes.

Spread three-fourths (about 3 cups or 778 grams) of cake batter into the prepared pan. Then spread half (about 1¾ cups or 359 grams) of the cheesecake mixture on top, leaving about a ½-inch border around the sides. Top with the remaining cake batter, and then top with the remaining cheesecake mixture.

Add the boiling water to a 13x9-inch baking dish or pan; place on the bottom rack of the oven. Place the springform pan on the middle rack of the oven.

Bake until the edges are set and the center is slightly jiggly, 65 to 70 minutes. Let the cake cool completely and then refrigerate it to set, at least 8 hours or overnight.

For topping: Combine cream cheese and butter in a bowl. Use an electric hand mixer to mix together on medium speed for 30 seconds and then add confectioners’ sugar and vanilla. Mix until combined and smooth, about 1 minute. Spread on top of cold cheesecake, and top with crushed chocolate cookies sandwiches, if desired. Refrigerate in an airtight container for up to 3 days. Best served cold.