Peel your potatoes and cut them into thick wedges. For medium size potatoes, cut them in half lengthwise and then cut each half into two wedges. Place your cut potatoes into a bowl of cold water.
Heat enough vegetable oil in a large deep pot or frying pan so that when you add the potatoes, they are covered.
Drain your potatoes very well and pat them dry. Add the potatoes carefully to your hot oil and cook over medium high heat for approximately 15 - 20 minutes, or until your potatoes are cooked through and golden.
When your potatoes are cooked, transfer them to a paper towel lined plate or colander to remove the excess oil and then toss them with the crumbled feta, dry oregano and salt. Set aside.
In a non-stick frying pan heat the olive oil. Then, crack one egg at a time into a small bowl and slip it into the pan and cook without flipping over. You can use a spoon to carefully pour some of the oil over the egg to help it cook. Your egg is ready when the circumference of egg white is browned and crispy; we prefer to keep our yolk somewhat runny.
Distribute your potatoes onto two plates and top each with a fried egg.
Top with additional salt and dry oregano if desired.
Serve immediately and enjoy!