Start with cleaning the underside of the bonnet, as you don't want dirt and water to drop down onto the engine bay, after you've cleaned it.
Spray down the underside of the bonnet with water via a hose or pressure washer. spray engine degreaser or all purpose cleaner diluted 4:1. I swear by Bilt Hamber Surfex HD.
Use the different size brushes/mitts solely for the wheels and not bodywork to agitate the **** off. Then hose down, and wipe with a microfibre cloth. If you have a leaf blower, use it to disperse the water for speed.
I personally don't spray the heat shield material on the underside of the bonnet. I just clean by spraying Surfex HD onto it and a damp microfiber cloth to avoid breaking the material up.
I then repeat the same method on the engine bay. However, I do not hose directly on the engine with a hose or pressure washer. It will be fine if you do, but I personally don't want to risk it, as the GT does not have a plastic cover like most newer cars. Instead I just use a spray bottle to get water and all purpose cleaner/engine bay degreaser onto the engine, agitate with brushes and wipe the dirty off with a microfibre.
If you want to be extra careful cut bits off a plastic bag to wrap and tape the electrical connectors to protect water ingress. You can also cover the alternator as well.
After the everything is wiped down and dried, I use Gyeon Tyre Shine on the plastics etc.
Shouldn't take more than 45min to do a decent job.
PS, the before photo was the way the previous owner left it. I wouldn't be so disgusting to leave the engine bay looking like that