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Preheat your oven to 350 degrees and grease an 8-inch square pan, and optionally line the pan with parchment paper.
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In a medium bowl, stir together the flour, baking powder, and salt and set aside. In a large bowl, beat the eggs with an electric mixer using a whisk attachment (you can use a hand mixer here or a stand mixer) until they are thickened. This should take about 3-5 minutes. Then slowly add in the sugar as you continue to mix until you have a very thick and fluffy mixture. It should ribbon back onto itself. Sift the flour mixture into the whipped eggs and gently fold until just combined.
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In a microwave safe bowl, melt your butter in the microwave, covering with a paper towel to ensure it doesn't splatter over. Then add in your whole milk, and heat in the microwave for about 30 seconds, or until it's just warm to the touch. If your mixture gets too hot, let it cool down slightly before adding it to the cake- it should be warm, not hot. Add in the vanilla extract, then stir the warm mixture into your cake batter until the mixture is just combined and there are no uneven streaks. Pour into your prepared cake pan and bake for 25-35 minutes or until a toothpick comes out clean. Cool completely in the pan.
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Use a toothpick or a fork to poke small holes all over the top of the cake all the way down the bottom of the pan. In a small bowl, stir together the espresso and vanilla extract. Brush the mixture over your cake generously- use the entire espresso mixture here. I like to brush it on slowly so that it soaks into the cake really evenly.