Line a 1.5–2 quart bowl with plastic wrap, leaving overhang. Pack in the vanilla ice cream, pressing firmly to eliminate gaps. Cover and freeze until solid, at least 4 hours (or overnight).
Preheat the oven to 350°F. Grease a 9-inch round cake pan, line the bottom with parchment, and lightly flour the sides; tap out excess.
Measure the granulated sugar and divide it into two portions: 0.5 cup for the cake and 1 cup for the meringue. Set both aside.
Make the sponge: In a heatproof bowl set over (not in) barely simmering water, whisk the large eggs with the reserved 0.5 cup sugar until lukewarm (about 100–110°F) and slightly thickened, 2–3 minutes. Remove from heat and beat with a mixer on high until pale and tripled in volume, 5–7 minutes. Sift the cake flour and fine salt over the foam in two additions, gently folding after each just until no streaks remain. Drizzle the unsalted butter around the bowl, add the vanilla extract, and fold in gently.
Spread the batter evenly in the prepared pan and bake until golden and springy, 12–15 minutes. Cool in the pan 10 minutes, then invert onto a rack, peel off parchment, and cool completely. If needed, trim the cake to roughly match the diameter of your ice cream dome.
Make the Swiss meringue: In a clean, grease-free metal bowl set over barely simmering water, whisk the large egg whites, the reserved 1 cup sugar, and the cream of tartar constantly until the mixture reaches 160°F and the sugar is fully dissolved, 3–5 minutes. Transfer to a mixer and whip on high speed until stiff, glossy peaks form and the bowl is cool, 5–7 minutes.
Assemble: Place the cake on a parchment-lined baking sheet. Unmold the ice cream onto the center of the cake and remove the plastic wrap. Working quickly, cover the ice cream and cake completely with the meringue, sealing it to the cake all around and creating swirls. Freeze to firm up, about 30 minutes.
Finish: Increase the oven temperature to 500°F. Bake the assembled baked Alaska until the meringue is evenly browned, 3–5 minutes, or brown it evenly with a kitchen torch, holding the flame 1–2 inches away. Slice with a warm knife and serve immediately.