Adjust racks to upper and lower-middle positions and heat oven to 350 degrees Fahrenheit.
Line 2 large baking sheets with parchment paper.
Whisk together flour, cocoa powder, baking soda and salt.
Melt semi-sweet chocolate chocolate in a heatproof bowl over a pan of simmering water. Stir until smooth and remove from heat. Allow cooling to room temperature but do not allow re-hardening.
Combine the eggs, espresso powder and pure vanilla extract. Set aside.
In the bowl of an electric mixer, beat the butter at medium speed until smooth and creamy. Beat in sugars until just combined, scraping down the sides of the bowl as necessary. With the mixer on low, gradually pour in the egg mixture and beat until incorporated. Add the chocolate in a steady stream and beat until combined. Scrape the bottom and sides of the bowl to make sure everything is incorporated. With the mixer on low, gradually add the dry ingredients and mix until just combined. Stir in peanut butter chips by hand. Do not over beat. Cover with plastic wrap and let rest for at least 2 hours.
Roll the dough into 2 oz balls and space about 2 inches apart on the baking sheet.
Bake until the edges of the cookies have begun to set but the centers are still very soft, about 14 minutes, rotating the sheets from front to back and top to bottom halfway through the baking time.
Remove from oven.
Cool the cookies on the sheets for 10 minutes, slide the parchment with the cookies onto wire racks, and cool to room temperature.