Linux

Well, for your particular case, you'd make a script, and a service to run that script on boot. Once the service starts, it will keep itself alive.

Here's the script:

bluetooth-reconnect.sh

#!/bin/bash

dbus-monitor --session "type='signal',interface='org.gnome.ScreenSaver'" |
  while read x; do
    if echo $x | grep -q "boolean false"
      bluetoothctl connect A1:11:22:3A:CD:F1
    fi
  done

You'd place this script somewhere that has system execution privilege (if your distro uses SELinux). I will use the directory /usr/scripts/ for example purposes (note that you will have to create this folder). Make sure to mark it executable with chmod +x /usr/scripts/bluetooth-reconnect.sh

You'd then write a service to start at boot, just really barebones and simple:

bluetooth-reconnect.service

[Unit]
Description=Reconnect Bluetooth after waking from sleep
After=default.target

[Service]
Type=simple
ExecStart=/usr/scripts/bluetooth-reconnect.sh

[Install]
WantedBy=multi-user.target

Move the service into /etc/systemd/system/ (filepath should be /etc/systemd/system/bluetooth-reconnect.service), and enable it and start it:

sudo systemctl enable bluetooth-reconnect.service && sudo systemctl start bluetooth-reconnect.service

And you should be good to go. At least assuming your distro doesn't have some specific quirk, which I wouldn't be able to help you with unless I knew what distro you run. Granted, this is my adaptation of what I saw in the linked forum and my own experience with services, I haven't actually tested this. But even if it has an issue, this will get you 90% of the way there, and there's a good chance it just works if the forum answers work for your distro.