“Now Big Wes wants you to kill a Mole rat Brood Mother, with one catch. You can only use pistols. Good luck piercing moleskin with those!”
Whack-a-Mole is an overseer quest and part of The Thrill of the Hunt multi-stage questline in Fallout Shelter.
Throughout the building you must clear several rooms of feral ghouls, radroaches, and mole rats. The brood mother is located in the first room on the top floor.