Combine flour, baking soda, and salt in a medium bowl, and set aside. In the bowl of an electric mixer fitted with the paddle attachment, cream butter and sugar on medium speed until light and fluffy. Add egg and vanilla; beat until well combined. Add peanut butter, and beat until combined.

Add the flour mixture all at once, and beat on low speed until just combined. beat until combined. Wrap dough in plastic wrap, and chill in refrigerator at least 2 hours. Preheat oven to 350 degrees. Line two baking sheets with parchment paper.

Pinch off about 2 tablespoons of dough, and make a well in your hand with the dough. Place one chunk of chocolate in the center, and enclose with the dough to cover completely. Roll dough into about a 1 3/4-inch ball with your hands.

Place the ball of dough on the baking sheet; repeat with the remaining dough and chocolate, placing cookies about 2 inches apart. Press 2 peanut halves into the top of each cookie.

Bake until cookies are golden, 16 to 18 minutes, rotating the pans halfway through. Remove from the oven; transfer cookies to a wire rack to cool slightly.