Preheat oven to 350°F. Line a 9"x9" baking pan with parchment paper, leaving 1 inch overhang on two opposite ends to serve as handles.
Combine the flour, oats, espresso powder, baking powder, and salt in a bowl and set aside.
Cream the butter, peanut butter, and sugars with an electric mixer until light and fluffy. Add the eggs and vanilla and beat until smooth.
Gradually add in the dry ingredients until just incorporated. The dough will be very stiff. Use a spatula to fold in about 3/4ths of the chocolate chips (by hand).
Dump the dough into the prepared pan and press evenly until level. Sprinkle the remaining chocolate chips over the top and gently press into the dough.
Place in the oven and bake for 30 to 35 minutes. Rotate the pan after 10 minutes. After 20 minutes, remove the pan from the oven and use an offset metal spatula to smear the melted chocolate chips around the top. Return to oven and bake for another 10 minutes. Check with a toothpick: inserted in the center, it should come out slightly moist with a few crumbs. If it needs more baking, tent with foil and bake another 5 minutes. Test again with the toothpick.
Remove from the oven and let cool for 15 minutes. Use the parchment paper handles to gently lift the entire square from the pan and place on a rack to cool completely before slicing.