Heat oil in a small saucepan to 375 degrees F (190 degrees C).
Spread bread crumbs into a shallow dish. Dip mushroom slices in beaten egg, dredge them in bread crumbs to coat, and cook them in hot oil until golden brown, about 5 to 6 minutes. Repeat until all mushrooms are cooked; drain on a plate lined with paper towels.
Blend feta cheese and lemon juice in a blender. Stream water into feta mixture as it blends until you get a desirable consistency. Serve with mushrooms.