What to say
People stall at this point. They're persuaded, they open the conversation, and then they stare at the empty box for ten minutes and close it again. So here's a plain way to write the message. None of it is research. It's what I think works, for reasons I'll give.
A good first message after a long gap has up to three parts.
Start with why you thought of them, and make it specific. "I walked past the sandwich place on Thursday." "They played that awful song from the Christmas party on the radio." "I finally read the book you kept telling me about." A specific prompt does two things. It explains why you're writing today and not some other day, so the message doesn't seem to come from nowhere. And it carries the real news, which is that they have a place in your head.
Then something brief about now. "How's the new job turned out?" "I hope your dad's doing better." One question is enough. Make it one they can answer in a line if they're busy.
The third part is optional, and it's a concrete suggestion. "Are you free for a coffee on Saturday the 14th? I could come to you." Or the setting up of a call, as in the last chapter.
Put together, that's three sentences.
"I walked past the sandwich place on Thursday and thought of you. How's the new job turned out? If you're free on Saturday the 14th I'd love to buy you a coffee and hear about it."
Here's what I'd leave out.
Leave out the long apology for the gap. Chapter 6 gave the reasons. "It's been far too long" is the most you need.
Leave out the account of your own news. It turns the message into a newsletter, and it can wait for the conversation.
Leave out anything that gives them a job. "We should really sort out a reunion with the old team" hands them a project. They'll mean to reply when they've thought about it, and they never will.
And above all leave out "we must catch up soon". That sentence has ended more friendships than any quarrel. It sounds like a plan and contains none. It lets both people feel that something has been arranged. The cure is a date. "Are you free on the 14th?" can be answered with yes, no, or "not then, but the 21st?". Any of those answers moves things on. "We must catch up" usually gets "definitely!", and then another fourteen months go by.
If you want to ask nothing of them, because you know they're stretched or grieving or ill, say so. "No need to reply, I just wanted you to know I was thinking of you." A message like that makes no demand, and people remember getting it.
When you do get to talk, don't stay in the shallows the whole time. Michael Kardas, Amit Kumar and Nicholas Epley ran a long series of experiments on conversation. They had people discuss either light topics or deeper and more personal ones, and predict beforehand how it would go. People expected the deeper conversations to be much more awkward than they turned out to be. They also underestimated how much the other person would care about what they had to say. After having both kinds, people felt more connected to the partner they'd had the deeper conversation with. Most of those pairs were strangers. The researchers found that with close friends and family people's predictions were more accurate, so the gap may be smaller with your friend from the old office. Still, after a long silence there's a pull towards keeping it all to holidays and house prices. If something real has happened to you in the fourteen months, say so. "It's been a rough year, to be honest. Mum's not been well." Usually the other person answers with something real of their own, and the conversation improves from there.
Lastly, there's the reply you give when they write back and you feel the old awkwardness about what to say next. Answer what they said, and move towards the date. You're not trying to be brilliant in writing. The aim is to end up in the same room.