Preheat the oven to 325°. Line 3 large baking sheets with parchment paper. In a medium bowl, whisk the flour with the baking powder and salt.
In a large bowl, using a handheld electric mixer, beat the butter until creamy. Add the confectioners' sugar and peanut butter and beat until light and fluffy. Add the vanilla extract. Beat in the dry ingredients at low speed. Scrape the dough onto 3 sheets of plastic wrap and pat into logs about 2 1/2-inches in diameter. Refrigerate the logs until firm.
Slice the logs 1/3 inch thick and arrange the cookies 1 inch apart on the baking sheets. Slightly flatten the cookies with your fingertips. Bake 1 sheet in the upper and 1 sheet in the lower third of the oven for 20 minutes, or until just golden; shift the sheets from top to bottom and front to back halfway through. Using a pestle or the bottom of a shot glass, gently make a 1 1/2-inch-round indentation in the center of each baked cookie. Let cool on the baking sheets. Repeat with the remaining cookies.
Meanwhile, put the chopped chocolate in a medium heatproof bowl. In a small saucepan, bring the cream to a boil: Pour it over the chocolate and let stand until melted, then whisk until smooth. Let the ganache cool to room temperature, then spoon it into the cookie indentations.
The cookies can be stored in an airtight container in the refrigerator for up to 1 week.