Some motherboards act that way. The BIOS Quick post setting can cause that type of boot issues. I've expericenced similar issues [strike]but can't recall if on or off causes it so try both settings.[/strike]
EDIT 1: The option is called 'Fast Boot' or 'Quick Boot' and should be enabled to quickly bypass the BIOS Post and go into Windows load.
1. On a motherboard I had that setting made the computer take about 3 minutes from the BIOS beep to start loading Windows 8.1. If I paused the BIOS post or entered the BIOS for a about a minute or even less, made it boot quick and normal, also pressing ctrl+alt+del to interrupt the boot between the beep and Windows load screen or as it got to the Windows load screen, made it reboot normally.
2. The same setting has caused similar issues in my present motherboard and old Dell computer I have... on that computer it can make the BIOS take longer to detect the IDE drives and disabling it makes the BIOS post go by in a few quick seconds.
3. So it's not a coincidence the setting can causes boot issues. If changing the setting doesn't make a difference, just optimize the BIOS configuring it to defaults failproof and/or optimal defaults.
4. You could also see if replacing the BIOS CR-2032 battery helps. Sometimes it's not providing the full 3 volts the BIOS needs to save settings, so it makes a recount on each restart. The battery is supposed to last a long time but it's not always the case, some motherboards come with a weak battery... I've answered to a few posts where this was the issue even though the motherboard was new or recently bought.
5. What I can't figure out is why unplugging the AC cord momentarily helps speed up booting... the only thing that comes to mind is capacitors may be having trouble retaining charge and somehow unplugging/replugging may cause a slight current spike boosting them or other motherboard elements.
Make Your PC Boot Faster With These BIOS Tweaks
http://www.pcworld.com/article/217052/speed_up_boot_time.html
EDIT 2: Or you could instead of shutting down, make Windows go into Sleep mode (not hibernate), So booting is not necessary. The computer should go from Sleep to the Desktop in a few seconds. This state needs AC power so don't cut it.
And if you need to cut power, set Windows to Hibernate... it boots from off to the desktop in 1/3 to 1/4th the time of a normal boot. This shuts the computer completely off and may not help avoid the double restart you have to do, but it may still boot faster. This requires an occasional normal boot to reset the Hiberfill.sys file data which can corrupt over time and may cause booting issues.
HIBERNATION VS. SLEEP
http://www.digitaltrends.com/computing/hibernation-vs-sleep-explained/