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Heat half the olive oil in a large frying pan and fry the shallot for 5 mins until softened. Tip in the mushrooms, turn up the heat, and fry for 5-10 mins until all the liquid released from the mushrooms has evaporated and they’re turning golden around the edges.
Add the crushed garlic and fry for another 30 seconds, then stir in most of the tarragon, the beans, crème fraîche, 1 tbsp water and some seasoning.
Use a spoon to make two dips in the mushroom and bean mix, and crack the eggs into them. Turn the heat to low and cover with a lid or a large baking sheet. Cook for about 6-8 mins until the egg whites are cooked and the yolks are still runny.
Meanwhile, toast the bread and rub lightly with the halved garlic clove. Brush or drizzle with the remaining oil. Top the mushrooms and beans with teaspoonfuls of the yogurt, then sprinkle over the reserved tarragon. Serve with the garlic bread.