Put an oven rack in the center of the oven. Preheat the oven to 375 degrees F. Line two baking sheets with parchment paper. Set aside.
In a medium bowl, sieve together the flour, cocoa powder, baking soda, and salt. Set aside.
In a stand mixer, fitted with the paddle attachment, beat the butter, peanut butter, 3/4 cup of the granulated sugar, and the light brown sugar together, scraping down the sides of the bowl as needed, until smooth, about 30 seconds. Add the egg and vanilla extract and mix until blended. With the machine running on low speed, gradually add the flour mixture until incorporated.
Put the remaining 1/4 cup granulated sugar in a small bowl. Form the dough into 1/4 cup-size balls and roll in the sugar. Arrange 5 balls of dough, evenly apart, on each baking sheet. Using a round 1/4 teaspoon measure, or the thin end of a wooden spoon, make a hole in the center of each ball of dough, 1/2 to 3/4-inch deep. Spoon 1 teaspoon of jam into each hole. Bake until the dough has spread and the surface of the cookies are crackled, 11 to 14 minutes. Cool for 5 minutes and then transfer to a wire rack to cool completely, about 20 minutes. Arrange the cookies on a serving dish and serve.
Recipe courtesy of Giada De Laurentiis