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Preheat oven to 350 degrees. Butter an 8-inch square baking pan or heatproof glass dish. Line pan with parchment paper, leaving a 1-inch overhang on 2 sides. Set aside. Place butter and chocolate in a large heatproof bowl set over (but not touching) simmering water; stir frequently until chocolate and butter are melted, about 7 minutes. Remove bowl from heat; let cool to room temperature, 10 to 15 minutes.
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Stir sugar into cooled chocolate mixture until combined. Whisk in eggs one at a time, whisking until smooth after each addition. Whisk in vanilla. Gently fold in flour and salt.
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Pour batter into prepared pan, and smooth top with an offset spatula. Bake until cake tester inserted in the center comes out with only a few moist crumbs attached, 40 to 45 minutes. Transfer pan to a wire rack to cool completely.
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Run a knife or offset spatula around the edges of the pan. Using parchment, lift brownies out of pan and onto the rack. Transfer to a cutting board; cut into 2-inch squares.