Heat the oven to 325°F and grease an 8 inch-square baking dish. Line the pan with parchment paper so there is a 2 inch overhang on opposite sides. These will serve as handles to lift the brownies from the pan.
Whisk together the melted butter, sugar, and cocoa powder in a large mixing bowl. Add the eggs and vanilla and vigorously whisk to combine.
Combine the flour, salt, and baking powder in a separate bowl, then stir it into the chocolate mixture until just combined. Set aside.
Add the butter, powdered sugar, peanut butter, and salt to a small bowl and mix until smooth.
Use the back of a spoon or an offset spatula to spread half of the brownie batter evenly into the baking dish. (The batter will be very thick).
Spoon half of the peanut butter swirl in tablespoon-sized dollops on top of the brownie batter.
Top with the remaining brownie batter and spread evenly, then dollop the rest of the peanut butter filling on top.
Insert a skewer or butter knife all the way to the bottom of the pan and drag the edge through the peanut butter dollops to make swirls.
Bake for 30-35 minutes, until puffed and lightly browned on top. Do not overbake.
A toothpick inserted into the center should come out a little wet with lots of crumbs.
Cool completely, then use the parchment handles to lift them from the pan and transfer to a cutting board.