Heat the oil with the butter in a frying pan over a high heat. Reduce the heat slightly, add the shallots and mushrooms and fry for 6 minutes, stirring occasionally, until the mushrooms are golden. Stir in the garlic and cook, stirring, for 1 minute.
Add the lemon juice to the mushroom mixture and season with salt and pepper.
Remove the pan from the heat and stir in the parsley. Keep warm while you poach the eggs.
Half-fill a separate frying pan with water and bring to a simmer. Break in the eggs and cook for 3 minutes.
Place 2 potato farls on each warmed serving plate and top with the mushroom mixture, then the eggs. Sprinkle with the chives and serve immediately.