When you moon see something, you're doing a moon viewing.
The readings for this word are both kun'yomi readings. For 見 you already learned the kun'yomi reading when you learned the kanji. For 月 you'll have to think back to the vocab word 月 which uses the つき reading. Combine these two together and you have つきみ.
October is moon-viewing season.
Proposing while looking at the moon sounds romantic.
An udon with an egg, please.