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Preheat oven to 350F. Lightly butter and line an 8-by-8-inch square baking pan with parchment paper or aluminum foil, allowing paper or foil to hang 2 inches over the edge. Lightly butter parchment. Set pan aside.
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Melt butter in a small pan set over medium heat. When it's melted, add sugar and stir until smooth and paste-like. Transfer to the bowl of a stand mixer fitted with the paddle attachment. Mix on medium speed until mixture has cooled so that it's no longer hot to touch (about 90-100F on an instant-read thermometer).
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With the mixer still running, add egg, yolk, and vanilla. Beat for 2-3 minutes, until mixture falls off the beater in thick ribbons.
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Reduce speed to low and add in flour, cocoa powder and salt, mixing until just barely incorporated. Scrape down sides of the bowl and with mixer again on low, add milk, beating until smooth.
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Pour into prepared pan, spreading into an even layer.
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Bake for 22-26 minutes, until a toothpick inserted into the center comes out with moist crumbs but no raw batter.
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Remove from oven and allow to cool slightly. If edges are sticking up a lot, while brownies are still hot, gently press down with an offset spatula to flatten.