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Step 1
Place the 1 stick unsalted butter for the cake in the bowl of a stand mixer and let it sit out at room temperature while you freeze the ice cream center. (Alternatively, use an electric hand mixer and large bowl.)
Step 2
Freeze the ice cream center: Coat a 3-quart glass or metal mixing bowl with cooking spray. Line the bowl with plastic wrap, leaving a 6-inch overhang around the edges. Fill the bowl with alternating scoops from 3 pints ice cream to create a mix of colors. (It’s OK if the ice cream melts slightly.) Smooth the top with an offset spatula and fold the plastic wrap over the top to completely cover. Freeze until completely firm, at least 2 hours. Meanwhile, make the cake.
Step 3
Make the cake: Arrange a rack in the middle of the oven and heat the oven to 350°F. Coat a 9-inch round cake pan with cooking spray and line the bottom with parchment paper. Add 1 cup granulated sugar to the bowl of butter and beat with the paddle attachment on medium-high speed until light and fluffy, 2 to 3 minutes. Add 2 large eggs and 1 teaspoon vanilla extract and continue beating until the mixture is smooth and lightened in color, 2 to 3 minutes.
Step 4
Turn the mixer off. Add 1 1/2 cups all-purpose flour, 1/2 teaspoon baking powder, and 1/4 teaspoon kosher salt. Mix on low speed until smooth, 1 to 2 minutes. With the mixer running, gradually add 1/2 cup whole milk and continue mixing until the batter is completely smooth, scraping down the sides halfway through mixing, 2 to 3 minutes total.
Step 5
Scrape the batter into the cake pan and spread into an even layer. Bake until the cake is lightly browned and a toothpick inserted into the center comes out clean, 25 to 30 minutes. Place the pan on a wire rack and let the cake cool completely in the pan.
Step 6
Make the meringue: Place 6 large egg whites in the clean bowl of a stand mixer fitted with the whisk attachment. (Alternatively, use an electric hand mixer and large bowl.) Add 1/4 teaspoon cream of tartar and whip on medium-high speed until foamy, 1 to 2 minutes. Increase the speed to high and gradually add 1 cup granulated sugar. Beat until the meringue is glossy and forms stiff peaks, 2 to 3 minutes.
Step 7
Assemble the baked Alaska: If browning with the broiler, arrange an oven rack 12 inches below the broiling element and set the oven to broil. Line a rimmed baking sheet or broiler-safe cake plate with aluminum foil. Invert the cake onto it and discard the parchment.
Step 8
Remove the bowl of ice cream from the freezer and uncover the top. Gently invert it onto the cake. Remove the bowl and plastic wrap. Use an offset spatula to spread a thick layer of meringue over the entire cake and ice cream dome. Use the tip of the offset spatula to create tall peaks in the meringue that will brown when torched.
Step 9
Toast the meringue: Broil just until the meringue begins to brown, 3 to 4 minutes. Alternatively, toast the meringue with a kitchen torch so the peaks are lightly browned. Transfer the baked Alaska onto a serving plate if needed, slice, and serve immediately.