Head into the Big Town common house and look for a death-obsessed young woman named Bittercup. You can either be rude or polite, but she'll ask you if you're seeing anyone if you express a common interest. Regardless of how you enter the conversation or how you proceed, she will hit on you.
If you tell her that no one understands you either, she will list every man she has dated in Big Town. After this, you can choose any of the first three responses and she will then want to start a relationship with you. Simply tell her you're not currently in any sort of relationship, and she will become your girlfriend. She will modestly end the conversation, but if you return to her once every 24 hours, she will give you one of the following presents each time:
The name of this quest is a play on the phrase "My cup runneth over", from Psalm 23:5 of the Bible, "Thou preparest a table before me in the presence of mine enemies: thou anointest my head with oil; my cup runneth over."