Cook the peas in a pan of boiling water for 2 minutes. Drain, then refresh under cold running water and drain again. Place in a food processor with the mint and blitz to form a rough texture. Add the butter, flour, yogurt, cheese and egg and season with salt and pepper. Blitz again to form a stiff paste.
Heat the oil in a large frying pan, then drop 2 tablespoons of the pea mixture into the pan for each pancake. Smooth the tops and cook over a medium heat for 3–4 minutes. Turn the pancakes over and cook for a further 2 minutes. Remove from the pan and keep warm. Repeat with the remaining mixture to make about 12 pancakes.
Heat a griddle pan until hot, add the broccoli and cook for 5–6 minutes, turning frequently, until slightly charred.
Top the pancakes with the poached eggs and serve with the charred broccoli.