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Preheat the oven to 350 degrees.
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Combine the butter and sugars in a large mixing bowl and beat with an electric mixer on medium speed until light and fluffy. Beat in the egg and the vanilla to combine.
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Stir together the graham cracker crumbs, flour, baking soda, and salt in a small bowl. With mixer on low, gradually beat that mixture into the butter-sugar mixture.
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Drop tablespoons of dough onto un-greased, unlined baking sheets and bake 8 minutes.
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Remove from the oven and quickly place a marshmallow half, cut-side down, onto each cookie. Return to the oven for 3 to 4 minutes, until light golden around the bottom edges.
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Remove from the oven and immediately press a square of chocolate onto each marshmallow.
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Let the cookies cool for 1 minute on the baking sheets, then carefully transfer to wire racks to cool completely. (A thin metal spatula is really helpful.)