Whisk the flour, baking powder, baking soda, and salt together in a bowl.
In your stand mixer (or use a hand mixer if you don't have a stand mixer) with the paddle attachment, cream the butter and both sugars together on medium speed until smooth, about 1-2 minutes. Add the eggs and beat on high until totally incorporated. Scrape down the sides and bottom of the bowl and then add the peanut butter and vanilla, then beat on high speed until combined.
Add the flour mixture to the peanut butter batter, then mix on low until just combined.
Chill the dough in the refrigerator for 30 minutes to an hour. Note: You can make dough and leave in fridge for a few days, just make sure to cover, and then to sit on counter for about an hour before scooping.
Preheat oven to 350°F. Line baking sheets with parchment paper or foil.
Scoop and roll cookie dough into balls. I use a regular ice cream scoop to scoop out, then divide each scoop in half and roll into balls. Now, I used a meat mallet that I only use for dough to make the indent on top, you can do that if you have a clean mallet, otherwise the traditional thing to do is use a fork to make a crisscross indent. Bake each batch for 13-15 minutes until very lightly browned on the sides. The centers will look very soft.
Remove from the oven. Cool cookies on the baking sheets for a few minutes before transferring to a wire rack to cool completely.
To melt the chocolate, place a saucepan with water about ⅓ full on stove and bring to a boil In a glass or metal small bowl put your milk chocolate (or dark if using). Place the chocolate bowl on top of the boiling water and melt chocolate (turn the flame to low so the water doesn't bubble up and splash into your chocolate), stirring every now and then until completely smooth and shiny. Dip about a ¼ of the cookie into the chocolate, drip off the excess, and place on parchment. Sprinkle each cookie right after dipping with the chopped peanuts so the peanuts adhere to the chocolate. Met the dark chocolate the same way and when melted, take a spoon and drizzle over the milk chocolate and chopped peanuts.