Grate the peeled potatoes on a box grater and add the grated potatoes to a large bowl filled with cold water. Strain the grated potatoes in a colander and rinse them under cold running water. Squeeze out as much liquid as possible with your hands, then wrap the grated potatoes in a cheesecloth and squeeze out as much liquid as possible.
Combine the grated potatoes, grated Cheddar cheese, 2 teaspoons of the salt and some pepper in a large bowl.
Combine the ketchup with the harissa paste in a small bowl and set aside.
Heat 1 tablespoon of the butter in a 6-inch skillet over medium heat until bubbling, 1 to 2 minutes. Add a quarter of the grated potato mixture to the skillet in a thin layer and cook until browned, 4 to 5 minutes per side. Transfer the cooked hash brown disk to a sheet tray.
Cook three more batches of the hash brown disks, adding 1 tablespoon of the butter and a quarter of the remaining grated potato mixture each time.
Cut a hole in the center of each of the hash brown disks with a 2-inch biscuit cutter.
Heat 1/2 tablespoon of the butter in the skillet over medium-low heat. Add 1 hash brown disk with a hole and crack 1 egg into the hole you created. Place a lid on top of the skillet to trap the steam and cook the eggs until they set, 2 to 3 minutes.
Gently transfer the Hash Brown Egg-in-a-Hole to a plate. Repeat with the remaining butter, hash brown disks and eggs in three batches.
Serve drizzled with the harissa ketchup.
Soaking the grated potatoes will help remove the starch.
If you don't have a cheesecloth, you can use a clean kitchen towel to squeeze liquid from the potatoes.
If you don't have a 2-inch biscuit cutter, use the cap of a Mason jar.