First make the homemade peanut butter chunks. Melt the white chocolate with the smooth peanut butter gently in a heatproof bowl set over a saucepan with simmering water.Spread it out onto a parchment-lined baking sheet, let it set until hard in the fridge and then just break it up into chunks.
Beat the butter with both sugars for 2 minutes on medium speed in the bowl of a stand mixer fitted with the paddle attachment or in a large bowl using a handheld electric mixer (or using a spatula and your own arm power!) until well combined and smooth, but not quite pale and fluffy. Mix in peanut butter, then beat in the egg. Add milk and vanilla and and mix until combined. Scrape down the bowl as needed.
Combine flour, baking soda and salt in a medium bowl and whisk to blend evenly. Add to the butter mixture and fold it in with a spatula until just combined. Add chocolate chips and most of the homemade peanut butter chunks (saving some for topping) and fold them through. Cover the bowl and place it in the fridge for 30 minutes.
Preheat oven to 350°F. Line 2 large baking sheets with parchment paper.
Scoop heaped tablespoon portions of dough and roll into smooth balls. Place them onto prepared baking sheets with 2 inches of space between each cookie. Flatten dough balls slightly and press a few extra peanut butter chunks on top.
Bake for 10-12 minutes until the surface is cracked and cookies are evenly golden. Let cookies cool on the baking sheet for 5 minutes before transferring them to a cooling rack.