Answer
See the explanation, which does not invoke momentum conservation.
Work Step by Step
The 2 pieces exert equal and opposite forces on each other, by Newton’s third law.
By Newton’s second law, the lighter piece experiences a larger acceleration ($a=\frac{F}{m}$), so it moves farther than the larger piece, during the interval that the force is acting.
When equal forces act on 2 bodies in the direction of their motion, the force that acts over a longer distance does more work.
The lighter piece has more work done on it, and by the work-energy theorem, ends up with more kinetic energy.
