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Preheat oven to 400°F with rack in middle. Butter and flour cake pan. Warm eggs (in shell) in hot water 5 minutes.
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Whisk together flour and salt.
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Beat eggs, ⅓ cup sugar, and vanilla with an electric mixer at high speed until tripled in volume and thick enough to form a wide flat ribbon that holds its shape on top of batter when beater is lifted, 7 to 8 minutes in a stand mixer or 14 to 16 with a handheld.
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Transfer to a wide bowl. Sift flour mixture, one third at a time, over batter, folding gently but thoroughly with a rubber spatula after each addition. Stir together butter and about ¼ cup batter in a small bowl until combined, then fold butter mixture into batter gently but thoroughly.
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Pour batter into cake pan and smooth top. Bake until golden brown and edges start to pull away from sides of pan, 12 to 14 minutes.
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Meanwhile, bring bourbon, water, and remaining tablespoon sugar to a boil, stirring until sugar has dissolved. Remove from heat.
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Transfer cake in pan to a rack, then run a thin knife between cake and side of pan and cool 5 minutes. Invert cake onto rack.