I'm writing a sci-fi & fantasy novel, I won't get into details, but in essence it's set on a fantasy planet that orbits around a sun, and has a moon. The basic premise (that I'm hoping for anyway) is that one side of the planet exclusively faces the sun and is in constant day-time, while the other face is submerged in darkness, with only the light of its moon to light it. What I'm wondering is, is it astronomically plausible for a planet to have one face be constantly sunlit, while the other is constantly moonlit?
At first, I considered the moon, which for some reason I had previously very incorrectly thought did not rotate around its own axis, and the same face was always in darkness/lit up. I'm now thankfully aware the moon does spin around its own axis, but the time it takes to orbit the Earth is practically equal to the time it takes to rotate around its own axis, keeping the same side of the moon facing the Earth throughout the month. If the moon didn't rotate around its axis/rotated at a different rate, different parts of the moon would face the Earth during the month.
In recap, I'm hoping for this planet to be plausible and definitely have the following criteria:
1. Have a moon and sun
2. Have the exact same face constantly facing its moon, and the other constantly facing its sun, so the exact same territory is always moonlit/sunlit, the boundaries don't change.
As long as these two criteria are generally met, I'm happy with any solutions/explanations for how this could be (if at all) astronomically/physically plausible.
Thanks in advance!
edit: i have made a rather large change, but don't want to change the question -- in essence, to see my temporary solution (still have to do some work on it) in progress, top answer's comments :)