Our grown-up ice cream cake is baked in a heatproof bowl, then hollowed out and filled with mint-chocolate-chip ice cream -- and finished with a crisp, shiny chocolate shell.
Preheat oven to 350 degrees. Brush bottom and sides of a large (approximately 10 inches in diameter and 5 inches deep) ovenproof glass, ceramic, or metal bowl with oil. Dust with cocoa, tapping out excess.
In a large bowl, whisk together cocoa, flour, sugar, baking soda, baking powder, and salt until thoroughly combined, 30 seconds. Add oil, eggs, buttermilk, warm water, and vanilla; whisk until smooth, about 2 minutes. Transfer to prepared bowl.
Bake until set and a tester inserted in center comes out clean, 1 hour, 25 minutes to1 hour, 45 minutes. Transfer bowl to a wire rack; let cool 15 minutes. Turn cake out onto rack; let cool completely.
Cut a 1-inch-thick slice from bottom of cake with a serrated knife. Invert "bowl" of cake; using a metal spoon or your fingers, hollow out all but a 1-inch border of cake (reserve scraps for snacking). Brush hollowed-out cake bowl and cut side of cake bottom with melted chocolate. Refrigerate until chocolate is firm, about 20 minutes.
Spoon ice cream into cake bowl, pressing to pack it all in and smoothing top. Return bottom of cake to bowl (it will be on top at this point). Cover with plastic; freeze until ice cream is firm,at least 4 hours and up to 3 days.
Unwrap cake and place, dome-side up, on a rimmed baking sheet lined with a wire rack. Place chocolate, butter, and corn syrup in a heatproof bowl set over (not in) a saucepan of simmering water. Stir occasionally until melted and smooth. Remove from heat and immediately pour over top center of cake, all at once. Shake baking sheet back and forth and tap it on a work surface a few times, if necessary, to ensure that glaze fully and evenly covers cake. Using two large, wide spatulas slipped underneath, transfer to a serving plate. Serve immediately, or freeze, uncovered, up to 1 day. Remove from freezer 10 to 15 minutes before serving. To serve, use a knife dipped in hot water, then wiped dry, to cut into wedges.
For a perfect dome shape, use Fat Daddio's 10-inch hemisphere cake pan (available at walmart.com).