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Preheat your oven to 350°F (175°C) and line baking sheets with parchment paper.
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In a saucepan over medium heat, melt the butter. Continue cooking, stirring frequently, until the butter foams and turns golden brown, about 5-7 minutes. Remove from heat and let cool slightly.
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In a large mixing bowl, combine the browned butter, brown sugar, and granulated sugar. Mix until smooth and well combined.
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Beat in the eggs one at a time, then add the vanilla extract, mixing until smooth.
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In a separate bowl, whisk together the flour, baking soda, and salt. Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
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Fold in the caramel bits and crushed cereal until evenly distributed throughout the dough.
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Drop rounded tablespoons of dough onto the prepared baking sheets, spacing them about 2 inches apart.
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Bake in the preheated oven for 10-12 minutes, or until the edges are golden brown.
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Allow the cookies to cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.
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Enjoy your delicious Caramel Crunch Brown Butter Cookies with a glass of milk or your favorite hot beverage!