Step 1

Sponge Cake: Preheat oven to 350°F. Brush a 10-inch round cake pan with oil. Line bottom with parchment; brush parchment. Whisk together flour, salt, and baking powder to combine. In a large heatproof bowl, whisk hot water with 1/4 cup granulated sugar until dissolved. Whisk in oil, then egg yolks and vanilla until smooth. Whisk flour mixture into oil mixture just until combined.

Step 2

In a mixer fitted with the whisk attachment, beat egg whites on medium-low speed until frothy. Add cream of tartar, increase speed to medium-high, and continue beating until soft peaks form. Gradually add remaining 2 tablespoons granulated sugar; continue beating until stiff, glossy peaks form, about 2 minutes. Stir one-third of egg whites into batter. Gently fold in remaining egg whites just until no streaks remain. Transfer to prepared pan, smoothing top with an offset spatula.

Step 3

Bake until cake is pale golden and top springs back when lightly pressed, 15 to 17 minutes. Let cool 15 minutes in pan on a wire rack, then turn cake out onto rack, remove parchment, and let cool completely.

Step 4

Cherry-Cookie Crumb: Pulse cookies, cherries, and granulated sugar in a food processor until finely ground. Add butter; pulse until texture resembles wet sand (you should have 2 packed cups). Transfer to a bowl, or refrigerate in an airtight container up to 1 week.

Step 5

Cream-Cheese Filling: Line a 10-inch-diameter mixing bowl (10-to-12-cup volume) with plastic wrap, leaving a long overhang on all sides. Beat cream cheese on medium-high until light and fluffy, about 2 minutes. Beat in condensed milk. In another bowl, whisk cream and vanilla to soft peaks. Fold into cream-cheese mixture to combine. Transfer to prepared mixing bowl, smoothing top with an offset spatula. Sprinkle evenly with 1 cup cookie-crumb mixture. Transfer to freezer.

Step 6

Cherry Ice Cream: Combine cherries, corn syrup, and cornstarch in a medium saucepan. Bring to a boil, stirring a few times. Boil until juices thicken slightly, about 1 minute. Transfer to a bowl and refrigerate until cool, about 30 minutes (or cool quickly over an ice-water bath). Temper ice cream in refrigerator until softened but not melting, about 20 minutes.

Step 7

Transfer ice cream and chilled cherry mixture to a food processor. Pulse just until combined and no streaks of ice cream remain. Pour into bowl in freezer, smoothing top. Top with sponge cake. Cover cake with plastic over-hang and freeze until firm, at least 8 hours and up to 5 days.

Step 8

Dip bowl in a hot-water bath 10 seconds. Wipe bowl dry and invert onto a cake plate. Remove bowl and plastic wrap. Using your hands, press remaining cookie-crumb mixture evenly over cream-cheese filling (remove any crumb mixture that falls onto plate with a pastry brush). Cut into wedges and serve immediately, or wrap in plastic and return to freezer until ready to serve, up to 3 days.