In a large skillet, heat butter and oil at medium heat. Once the butter has melted, add the onions and stir gently to coat them with the butter and oil. Arrange the onions evenly in the skillet. Reduce the heat so that the onions do not burn.
Allow the onions to cook, stirring every 5 to 10 minutes until they are soft and acquire a deep brown color (about 40 minutes).
Add garlic and sun-dried tomatoes and stir for about 1-3 minutes or until fragrant. Arrange the mixture evenly in the skillet and then carefully crack the eggs on top.
Sprinkle with crumbled feta, salt and pepper. Cover with a tight fitting lid and cook for 10-15 minutes.
Remove the skillet from the heat, garnish with chopped parsley and serve.