Preheat oven to 350°F (177°C). Line two large baking sheets with parchment paper or silicone baking mats. (Always recommended for cookies.) Set aside.

Toss the flour, salt, baking soda, and baking powder together in a large bowl. Set aside.

In a medium bowl, whisk the melted butter, granulated sugar, and brown sugar together until no brown sugar lumps remain. Whisk in the egg, then the peanut butter until combined. Finally, whisk in the vanilla. Pour the wet ingredients into the dry ingredients and mix together with a large spoon or rubber spatula. The dough will be very soft and slick. Fold in the oats and the 1 and 1/4 cups chopped peanut butter cups, reserving the remaining 1/4 cup for later.

Roll the dough into balls, about 3 Tablespoons of dough each. I highly suggest using a large cookie scoop to make it easier. Place 8 balls of dough onto each cookie sheet.

Bake the cookies for 11-12 minutes. The cookies will look very soft and underbaked. They will continue to bake on the cookie sheet. Remove from the oven and lightly press down on each cookie to slightly flatten, since the cookies will not spread all the way in the oven. Press a piece or two of the remaining chopped peanut butter cups into each cookie. Allow to cool on the cookie sheet for 10 minutes before moving to a wire rack to cool completely.