Preheat oven or air fryer to 400℉. Cut potatoes into small cubes and soak in warm water for 10 minutes. Drain and toss in 1 teaspoon of olive oil. Cook for 15-20 minutes or until fully cooked and slightly crispy. Remove from air fryer and set aside.
In a large skillet, add 1 teaspoon of olive oil over medium heat. Add in the ground beef and break up the beef as it cooks. Continue to cover over medium heat until the ground beef is fully cooked. About 5 minutes. Drain excess fat and season with salt, pepper, garlic powder, and cayenne pepper. Remove the beef from the pan and set aside.
In the same pan add in minced shallots and sliced mushrooms. Season with salt and pepper and cook for 5-6 minutes or until the mushrooms begin to brown. Mix in the ground beef, cooked potatoes, and Greek yogurt.
In another pan add in 1/2 teaspoon olive oil over medium heat. Crack the eggs and cook until the whites are set and flip. Cook for another 30 seconds if you would like over easy eggs. For over medium eggs cook for another 20-30 seconds and for scrambled eggs stir the eggs to break apart. Continue stirring until the eggs are fully cooked.
Spoon the beef mixture into bowls and add in the shredded cheese and eggs. Top with avocado chunks, and more Greek yogurt if desired.