So I've been searching around and I'm having trouble finding what is written on the letters Byleth receives on their birthday, namely those that come after the time-skip.
From what I could gather there are 5 letters post time-skip:
AM: Dimitri or Gilbert, depending on the month (I found these ones on Reddit);
CF: Edelgard;
SS: Flayn or Seteth, I saw conflicting reports so I'm not sure;
VW: Claude.
For now, I'm leaving the letters I could find here for future reference and I'll add the missing ones as I find them. If you know any a mistake let me know!
EDIT: So I remembered to look for a save editor for the game and voilá, I added all the available letters. Strangely enough, there's a letter from Rhea that I don't think it's available in-game, so that's definitely interesting. This should be all the letters and they should be in the correct order, but you spot any mistake let me know!
Pre Time-Skip:
Claude: "Happy birthday, Teach! You're always working so hard, we got you this so you'd stop that. Kidding, of course!"
Dimitri: "Happy birthday, Professor. It isn't much, but we got you a gift as a symbol of our gratitude. I hope you will accept it."
Edelgard: "I hear that it's your birthday, Professor. I hope you have a joyous day. Here's a gift from the house...myself included."
Post Time-Skip
Claude (VW): "They say it's your birthday. Is that true? Either way, your Golden Deer got you a little gift. Enjoy!"
Dimitri (AM): "As trite as it sounds, I wish to thank you. For everything. I hope you have a lovely birthday, Professor".
Edelgard (CF): "It's your birthday, is it not? I wish you a truly happy day. Even with times as they are, won't you celebrate with us?"
Flayn (SS): "On behalf of all, I wish you a most joyous day of birth! I ask that you spend it in absolute tranquility."
Gilbert (AM): "We should all be grateful that you were born and found your way here. I mean that."
Rhea (Unused?): "Today is the day you were born into this world. In the name of the goddess, I wish nothing but blessings upon you."