Preheat oven to 350°F / 176°C with racks in the top and bottom third.
Line two baking sheets with parchment paper.
Beat the butter until fluffy in the large bowl of an electric mixer. Add the sugars, and mix until creamy, scraping the sides of the bowl once or twice along the way. Mix in the peanut butter, the eggs one at a time, and then the vanilla extract.
Combine the flour, baking powder, baking soda, and salt in a medium bowl. Stir well, and add to the wet ingredients in two batches, mixing on low just until there are no dry patches. Stir the oats and peanuts in by hand until just incorporated.
Use a spoon or scoop to shape the dough into balls using about 3 tablespoons for each cookie. Arrange on the baking sheet separated by at least 1 ½-inches. Chill for 30 minutes in the refrigerator.
Remove from refrigerator, and press each dough ball twice with the tines of a fork creating a grid pattern. Bake for 12 minutes or so, rotating sheets halfway through. They should be puffed, golden at the edges and on any peaks.
Cool cookies on a rack and enjoy!
