You can include «мне» in Russian. But it should be «мне» (Dative case), not «меня» (Accusative case; here, «яйцо» is this case).
I'm not sure why this exact translation has «me» in English but doesn't have it in Russian. Maybe it would sound unnatural in English without «me»?
Could you pass the salt to me, please? Could you pass me the salt, please? Could you pass the salt, please? Pass the salt, please. Salt, please.
All sound perfectly okay. If it were me speaking, I would use any of the shortest 3 variants for brevity. The longer ones carry a tone of being more polite. But they all still sound perfectly fine.