Line two half sheet pans with parchment paper or wax paper then set aside.
Add the peanut butter, softened butter, and vanilla extract to the bowl of an electric mixer, or to a large mixing bowl if using a handheld mixer, then beat on medium-high speed until creamy, 30 seconds.
Add 2 cups powdered sugar, 1 cup at a time, beating on medium speed until combined between additions. Scrape down the sides of the bowl and the paddle attachment if using then give the ingredients a good mix with a spatula to make sure there isn’t any powdered sugar stuck at the bottom of the mixing bowl. Add the remaining 1 cup powdered sugar 1/2 cup at a time, beating until well combined between additions. The dough will be very thick.
Scoop the dough out by the Tablespoon then roll into a ball and place onto the prepared sheet pans. If the dough becomes sticky, you can stick it in the freezer for 10 minutes. Once all the dough has been rolled into balls, place the sheet pans into the freezer to harden for 30 minutes.
Create a double boiler by adding a couple inches of water to a small saucepan then bringing it to a simmer. Place a heat-resistant bowl like a glass bowl or stainless steel mixing bowl on top, ensuring the bottom of the bowl doesn’t touch the water, then add the chocolate chips and oil to the bowl. Turn the heat down to low then stir the chocolate chips with a spatula until they’re melted and smooth.
Working with one sheet pan of peanut butter balls at a time (leave the other one in the freezer), stick a toothpick in the center of a ball then dip it into the melted chocolate, leaving the top of the peanut butter ball exposed. Don't stick the tooth pick in too far otherwise it will be hard to remove. Gently spin the toothpick between your fingers to remove any excess chocolate then place the peanut butter ball back onto the baking sheet. Use your finger to gently press it off the toothpick. Repeat with remaining peanut butter balls then use your fingertip or a toothpick to smooth out the holes on top of the Buckeyes. Refrigerate for 15-20 minutes to set the chocolate.
The Buckeyes are fine to sit out at room temperature for 24 hours on a cookie plate for example. Store in an airtight container in the refrigerator for up to 2 weeks when not eating.
Notes
A drippy/runny natural peanut butter will not work for this recipe. I like Jif Naturals Creamy Peanut Butter which, despite the name, is not runny/drippy.
Buckeyes freeze very well. Place the sheet pans with the Buckeyes into the freezer until frozen solid. Transfer to a gallon Ziplock freezer bag then freeze for up to 3 months.