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Preheat the oven to 350°F with a rack in the center. Line a 9" x 13" baking pan with parchment, leaving a slight overhang for easy removal, and lightly grease the parchment.
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To make the batter: In a large bowl or the bowl of a stand mixer fitted with the flat beater, add the eggs, cocoa, peanut butter, vanilla, salt, baking powder, and espresso powder. Beat to form a thick, uniform paste, scraping the bowl as needed.
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In a saucepan set over low heat (or a medium microwave-safe bowl), melt the butter, then add the sugar and stir to combine. Continue to heat (or microwave) briefly, just until the mixture is hot but not bubbling (about 110°F to 120°F); it’ll become shiny as you stir. Add to the egg mixture and stir until smooth and shiny.
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Add the flour and stir just until a few dry streaks remain.
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Add the peanut butter chips and stir until well combined.
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Transfer the batter to the prepared pan and smooth into an even layer. It will be slightly thicker than some brownie batters; that’s OK.
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To make the peanut butter swirl: In a small bowl, stir the peanut butter and sugar to combine (the sugar will not dissolve). Dollop the mixture on top of the brownie batter and use a butter knife or wooden skewer to swirl it gently, creating a marbled effect. Sprinkle with additional peanut butter chips if desired.
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Bake the peanut butter brownies for 28 to 32 minutes, until the edges feel set and a toothpick or the tip of a sharp knife inserted into the center comes out with moist crumbs clinging to it but no raw batter. A digital thermometer inserted in the center should register 205°F when the brownies are done.
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Remove the brownies from the oven and let cool in the pan on a rack. Use the parchment to lift the brownies out of the pan and transfer them to a cutting board for slicing.
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Storage information: Store leftover peanut butter brownies in an airtight container at room temperature for several days; freeze for longer storage.
