It needs to be faster, more like 3.4 km/s

Throwing an object at the escape velocity just means the gravity of the moon will slow the object down until that object is not moving relative to the moon. The moon orbits the earth at 1.02 km/s, so the object would end up orbiting the earth.

Throw something (2.38 + 1.02) km/s in the opposite direction of the moons orbit and it will end up with an orbital speed of 0, causing it to fall down to earth.