Preheat oven to 375 degrees. Line baking sheets with silicone mats or parchment paper.
In a medium bowl whisk together flour, cocoa, baking soda, and salt. In a separate bowl using an electric mixer, beat the sugars, butter, and 1/4 cup of the peanut butter until light and fluffy. Add the vanilla and egg, beating to combine. Stir in the flour mixture, combine well. Set dough aside.
In a small bowl stir together the powdered sugar and remaining 3/4 cup of peanut butter until smooth.
Break of tablespoon-sized pieces of dough and flatten with your hands. Spoon about 2 teaspoons of peanut butter into the center of each dough round. Wrap the chocolate dough around peanut butter center, pressing to seal. Place dough rounds about 2 inches apart on prepared baking sheets. Flatten dough to 1/2-inch thickness with the bottom of a flat spatula or glass. Bake 7-9 minutes, or until set.