I was just happy to see that there was part of the moon I couldn't even see with my own eyes! That amazed me.
You can actually see the bit of the moon that's illuminated by Earthshine quite easily with the naked eye at times, if you know when to look. There are three criteria that need to be fulfilled, and they're all fairly obvious if you think about it, but their implications can be subtle.
1. You need the sky to be fully dark, so that you can distinguish the faint Earthshine from the surrounding sky. That means at least an hour after sunset (maybe 2 hours after sunset in the summer, because the sun travels below the horizon more slowly at that time of the year) and at least an hour before sunrise (2 hours in summer).
2. You want the moon to be a fairly slender crescent, so that the brightness of the sunlit bit doesn't overwhelm the Earthshine. Idealy that means about 3 to 7 days after new moon, when the moon is visible in the west after sunset, or about 3 to 7 days before new moon, when the moon is visible in the east before sunrise.
3. (This one is a bit more subtle.) You want the Earthshine to be as bright as possible. Since Earthshine is reflected sunlight, you want the hemisphere which is opposite the moon - ie the hemisphere which would be visible to an observer on the moon - to be fully illuminated by the sun. However, you're in that hemisphere, by definition since you can see the moon, and we've already seen that you want it to be dark where you are. So basically that means you want the moon to be close to the horizon, so that as much as possible of that hemisphere (to the west after the sun has set where you are, or to the east before the sun has risen where you are) is still daylit.
Put all that lot together, and assuming you don't want to be up at stupid o'clock in the morning, and it means that you want to be looking for Earthshine about 1-2 hours after sunset, around 3-7 days after new moon. Around 9pm tonight should be good.