Prepare and preheat a charcoal grill for indirect cooking, with one hot side and one hold (unheated) side.
In a large bowl, combine the tomato, avocado, ½ cup of the olive oil, the scallions, olives, vinegar and oregano. Season it with a pinch of salt and twist of pepper and toss to combine.
Set a large cast iron skillet on the hot side of the grill to heat for 1 minute. Wrap the pita in foil.
Add the remaining 3 tablespoons of olive oil to the skillet and heat until shimmering. Scatter the feta evenly across the bottom of the skillet and cook, without stirring, until slightly melted and lightly crisp on the edges, about 1 minute. Crack the eggs into the skillet and cook, without stirring, until a crust begins to form, about 1 minute. Place the pita on the hold side of the grill. Cover the grill and cook until the whites are set and the yolks are slightly runny, about 1 minute more.
Remove the skillet and pita bread from the grill. Transfer the eggs to a platter, top them with the tomato salad and serve with warm pita.