Preheat oven to 350°F. Line two baking sheets with parchment paper.
In a large bowl, combine flour, baking soda, salt, and oats. Set aside.
In a large bowl, with a hand mixer or stand mixer fitted with the paddle attachment, beat butter and peanut butter until smooth. Add both sugars and beat until combined. Beat in egg and vanilla, scraping the sides of the bowl as needed. Add flour mixture and mix on low speed until thoroughly combined. Stir in ½ cup chocolate/peanut butter chips, reserving the rest to add on top of the cookies.
Scoop the dough** into generous 2-tablespoon balls and place the dough balls a couple inches apart on the prepared baking sheets. Flatten the dough mounds just slightly. Using the remaining ¼ cup chocolate chips, press several chocolate chips onto the tops of each dough mound (this makes for a prettier cookie).
Bake for 10-13 minutes, until the edges are just beginning to darken, but the centers are still underdone (the cookies should be so soft/underbaked that you can't pick them up yet). Place the baking sheets on wire racks to cool and firm up. Sprinkle with a flaky finishing salt, if desired. Repeat with remaining dough allowing the baking sheets to cool completely between batches.
Notes
*I prefer an emulsified/no stir peanut butter for this recipe because it produces the most consistent result. You can use an all-natural (separated) peanut butter, but just be aware that depending on the oil content the cookie texture might be slightly different than those pictured. Also, make sure you stir it really well.**If you prefer thicker cookies, chill the dough for a couple hours before baking (you'll likely need to add a minute or two to bake time). Note: The cookies pictured are from dough that has not been chilled. If you scoop your dough into balls that are more or less than 2 tablespoons, be sure to adjust the bake time accordingly (likely 1-2 minutes in either direction).