Preheat oven to 350F (180C). Line an 8×8 inch pan with parchment paper in a crisscross fashion so that all 4 sides of the pan are lined with parchment paper.
Sift flour and cocoa powder into a medium mixing bowl using a fine mesh sieve.
Melt chocolate, butter and oil over a double boiler or in the microwave at 30 second intervals. Mixture should be stirred intermittently to prevent chocolate from burning. Once all of the chocolate is melted, set chocolate mixture aside to cool slightly before using, about 8-10 minutes.
In a small mixing bowl, add peanut butter and heat up in the microwave for 20-30 seconds to soften into a pourable consistency. Whisk in the powdered sugar into peanut butter and set aside until ready to use.
In a stand mixer fitted with paddle attachment or with a hand-held electric mixer, beat the eggs and sugar together at medium speed until mixture doubles in size and takes on a pale-yellow thick consistency, about 4-5 minutes.
Mix vanilla extract into wet batter followed by the cooled chocolate mixture until thoroughly combined, about 1-2 minutes.
Using a flexible spatula, fold in salt, sifted flour and cocoa powder into wet batter just until brownie batter comes together without any streaks of unmixed flour. Take care not to over mix brownie batter.
Transfer brownie batter into prepared pan and smooth into an even layer using the back of a rubber spatula.
Spoon peanut butter mixture (about 1/2 tablespoon dollops) over the top of brownie batter and use a butter knife or skewer to swirl dollops of peanut butter into brownie batter.
Bake peanut butter swirl brownie batter for 25-30 minutes. Brownies are ready when a toothpick inserted into the center comes out clean with a few moist crumbs. I find that the 27-28 minutes mark yields the perfect fudgy brownie consistency.
Cool brownies in pan for at least 30 minutes before cutting and serving.
Notes
Peanut butter swirl brownies are best eaten the day it is baked; however, leftover brownies stored in an airtight container will last 2-3 days at room temperature and 1 week in the fridge. Personally, I recommend storing brownies in the refrigerator to prevent them from drying out. Peanut butter swirl brownies can be easily heated in the microwave or toaster oven to serve warm.
Peanut butter swirl brownies stored in an airtight, freezer safe container will last up to 1-2 months stored in the freezer. When ready to eat, simply thaw in the refrigerator overnight. To serve warm, brownies may be heated through in the microwave or toaster oven.
Allow peanut butter swirl brownies to rest in pan for at least 30 minutes once baked. This will make it easier to cut into desired shape and serve. Additionally, using a shape knife will help in getting clean even cuts. I like to wipe the knife in between cuts with a clean kitchen towel.