Most Helpful Guy
Nope -- and I actually recommend strongly against this. I did write a letter once, horrible idea in retrospect.
Confessing feelings when someone else is trying to distance themselves away from you generally doesn't bring them back. It might even push them away further. And it makes the break up more awkward than it needs to be.
If you want someone back, it's generally better to attract them back to you by being attractive to them and giving them a chance to miss you. It's kind of like the reset button was pushed on the relationship. Just like it's not good to confess your deepest feelings to a crush you never dated, it's generally also not so good to confess your deepest feelings to someone who broke up with you. It might be reasonable to ask them out at some point but it's generally better to keep it light and easy and friendly.