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Preheat the oven to 325°F. Lightly grease an 8" round pan that’s at least 2" deep, and line the bottom with a parchment round or a silicone liner; grease the parchment or liner.
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In the bowl of a stand mixer fitted with the whisk attachment, or in a large bowl using a handheld electric mixer, beat the eggs, sugar, and vanilla until thickened and light gold in color, about 3 to 4 minutes on medium-high speed. The batter should fall in thick ribbons from the whisk or beaters.
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With the mixer still on medium-high speed, gradually add the oil, beating until incorporated.
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Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Sift the flour, cocoa, baking powder, and salt into the egg mixture and beat just until combined. Scrape the bottom and sides of the bowl, then beat again to incorporate.
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In a saucepan set over medium heat, bring the milk, chocolate, butter, and espresso powder to a rolling boil, stirring often. Remove the pan from the heat and stir until the chocolate is completely melted, 1 to 2 minutes.
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With the mixer on medium-low speed, gradually add the hot milk mixture to the batter, mixing until well combined. Scrape the bowl, then mix briefly, just until the batter is smooth.
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Pour the batter into the prepared pan.
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Bake the cake for 40 to 45 minutes, until the edges are beginning to pull away from the sides of the pan, a toothpick inserted into the center comes out clean, and the top feels set.
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Remove the cake from the oven, carefully loosen the edges, and let cool in the pan for 20 minutes. Turn the cake out onto a wire rack and remove the parchment round. Turn the cake right-side up and let cool to room temperature before frosting. (Use it in our Giant Hostess Cupcake Cake or frost it with Whipped Caramel Frosting.)
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Store the frosted cake, covered, at room temperature for up to 4 days. The unfrosted cake can be frozen for up to 1 month.
