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Melt the butter in a saucepan over medium heat and cook until it turns golden brown with a nutty aroma, then remove from heat and let it cool slightly.
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Pour the browned butter into a bowl and mix in the brown sugar and white sugar until smooth.
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Add the egg and vanilla extract, then stir until well combined.
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In a separate bowl, mix the flour, baking soda, and salt.
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Add the dry ingredients to the wet mixture and mix gently to form a soft dough.
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Heat the oven to 350°F (175°C).
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Scoop the dough onto a lined baking sheet, leaving space between each portion.
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Bake for 10 to 12 minutes until the edges are set and lightly golden.
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Let the cookies cool for a few minutes before serving.