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Preheat the oven to 350°F. Lightly grease (or line with parchment) two baking sheets.
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To make the cookies: Combine the sugars, peanut butter, shortening, salt, baking soda, and vanilla, beating until smooth.
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Add the egg, beating to combine.
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Stir in the oats and flour.
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Drop the dough by the heaping teaspoonful onto the prepared baking sheets, leaving about 1 1/2" between them. A teaspoon cookie scoop works well here.
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Using the flat bottom of a drinking glass dipped in sugar, flatten each cookie to about 1/4" thickness, about 1 1/2" diameter. If you're a fan of the popular sweet/salty flavor combo, sprinkle each cookie with the tiniest amount of extra-fine salt (or to taste; you know your own tolerance for saltiness).
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Bake the cookies for about 12 to 14 minutes, until they're a medium golden brown.
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Remove them from the oven, and transfer them to a rack to cool. Once they're cool, wrap airtight; they'll soften if left unwrapped.
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Decide which type of filling you want to use. Each recipe makes enough to fill all the cookies.
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To make the peanut butter filling: Mix all of the ingredients until smooth; a mini food processor works well here. If you're not using a food processor, sift the confectioners' sugar before stirring it together with the peanut butter and milk or half-and-half.
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To make the fudge filling: Combine the chocolate chips and half-and-half in a small, microwave-safe bowl; or in a small saucepan. Heat until the half-and-half is steaming. Remove from the heat, and stir until the chips have melted and the filling is smooth; reheat briefly if necessary.
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Spread the filling on half the cookies. Top with the remaining cookies.