Sift together 1 cup flour, 3/4 cup unsweetened cocoa powder, 1 teaspoon baking soda and 1/4 teaspoon salt. Set aside.
In a stand mixer fitted with a paddle attachment or a large mixing bowl with electric hand mixer, cream the softened 1/2 cup unsalted butter, with 2/3 cup sugar and 1/4 cup light brown sugar, until light and fluffy. Add the 1 large egg, and 1/2 teaspoon vanilla. Blend until smooth.
Slowly add dry ingredients to the wet ingredients until just combined.
By hand, fold in the 1 cup peanut butter chips. If you do this with the electric mixer, the chips with break into smaller pieces because the dough is so dense.
Divide the dough in half. Form each half into a log about 1 1/2 inches in diameter. Roll in plastic wrap and refrigerate for at least an hour.
Preheat oven to 350°F. Remove the cookie dough from plastic wrap and slice into 1/2 inch thick medallions. Place on a cookie sheet lined with parchment paper.
Bake for 10 minutes. They might look slightly undercooked, but will harden up while cooling.
Allow to cool for 5 minutes before removing to a cooling rack. Repeat with the remaining dough.
For Creamy Peanut Butter Frosting:
In a stand mixer fitted with a paddle attachment or a large mixing bowl with electric hand mixer whip the softened 6 tablespoons butter until pale and fluffy. Add the 1/2 cup creamy peanut butter, incorporating fully.
Add the 2 cups powdered sugar. Mixture will be dry and grainy.
Stir in the 1 tablespoons milk and 1/2 teaspoon vanilla, beating until smooth.
Place heaping tablespoons of creamy peanut butter frosting on the inside of one cookie and top with another to form a cookie sandwich.
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