Preheat oven to 350°F (180°C). Line 2 rimmed baking sheets with parchment paper.
In the bowl of a stand mixer fitted with the paddle attachment, beat butter, peanut butter, and sugars at medium speed until creamy, 1 to 2 minutes, stopping to scrape paddle and bottom and sides of bowl. Add egg and egg yolk, beating until combined. Beat in vanilla; scrape sides of bowl.
In a medium bowl, whisk together flour, oats, salt, baking powder, and baking soda. With mixer on low speed, gradually add flour mixture to butter mixture, beating just until combined and stopping to scrape sides of bowl. Fold in chocolate pieces and white chocolate.
Using a 3-tablespoon spring-loaded scoop, scoop dough (about 60 grams each), and place 2 inches apart on prepared pans.
Bake, one pan at a time, until edges are golden brown and set, 12 to 15 minutes. (Second pan can stand at room temperature while the first bakes.) Immediately smash cookies with bottom of a glass or jar. Garnish with chocolate pieces and white chocolate, if desired. Let cool completely on pans. Store in an airtight container for up to 3 days.