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Peanut butter cookies: Sift together the flour, baking soda, and baking powder in a small bowl. Stir in the oats and salt. Set aside.
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Combine the butter, peanut butter, and sugars in the bowl of a stand mixer fit with the paddle attachment. Cream on medium speed until very fluffy, 3 to 4 minutes. Stop the mixer as necessary to scrape the sides and bottom of the bowl.
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Turn the mixer to medium-low and add the eggs one at a time, beating well after each addition. Beat in vanilla extract. Turn mixer to low and add the dry ingredients, mixing until just incorporated. Scrape down the sides and bottom of the bowl again to ensure the ingredients are well combined.
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Line 2 large baking sheets with parchment paper and drop heaping tablespoons of dough onto lined sheets. (You should have about 48 cookies.) Do not smoosh the cookies down; they will flatten as they bake. Refrigerate the scooped cookies for 2 hours.
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Preheat the oven to 375 degrees. Arrange 8 cookies on a parchment-lined baking sheet, leaving the rest of the cookies chilling in the fridge while they bake. Bake until golden, about 12 minutes. Place baking sheet on a wire rack to cool for 10 minutes, then remove cookies to rack to cool completely. Repeat this process with remaining cookies. Allow the cookies to cool completely before filling them.
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Chocolate peanut butter filling: Place heavy cream in a small saucepan and heat until it’s just steaming. Place chocolate and peanut butter in a small mixing bowl and pour the heavy cream over it. Allow to sit for 3 minutes, then whisk until chocolate is melted and mixture is smooth. Refrigerate until thickened, about 1 hour.
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Beat cooled ganache with an electric mixer set to low for 2 minutes. Gradually increase the speed of the mixer to medium-high. Ganache is ready to use once it is light and fluffy.
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Spread or pipe filling onto half of the cookies and top each with another cookie. Serve and enjoy. (I recommend storing them in the freezer. They taste even better when they’re frozen solid.)