In the bowl of a stand mixer fitted with a paddle attachment, beat together peanut butter and sugar until well combined. Scrape down as needed to ensure thorough mixing. Add egg and mix until incorporated.
In a large bowl, whisk together flour, baking powder, baking soda, and salt. Add dry mixture to mixing bowl in three additions, alternating with buttermilk. Start and finish with dry mixture. Mix on low speed until dough has formed and there are no longer any dry streaks of flour.
Scrape out dough and place on a parchment lined baking sheet. Roll out or pat dough to about 3/4-inch thickness. Cover with plastic wrap and let chill in the freezer for 15 minutes.
Meanwhile, add about 3-inches of vegetable oil (or shortening) to a heavy bottomed pot (Dutch oven and cast iron pans work best). Attach candy thermometer to pan. Set over medium heat until oil reaches 350°F.
Remove chilled dough from freezer. Use a floured 3 1/2-inch doughnut cutter (or 3 1/2-inch round cutter and 1-inch round cutter) to punch out as many doughnut rings and doughnuts holes as you can. Gather up the scrape and re-roll dough up to two times to punch out additional rings and holes.
Carefully place two or three doughnut rings into hot oil. Do not overcrowd the pot. Cook for about 90 seconds on each side, until golden brown. Cook doughnut holes for about 55 seconds on each side. Use a slotted spoon to transfer cooked doughnuts onto a paper-towel lined baking sheet (or to a wire rack over a baking sheet). NOTE: Oil temperature will change as you add raw doughnuts into the pot. Monitor oil temperature using a kitchen/candy thermometer to make sure doughnuts do not burn.*