Preheat the oven to 200c/400f and put your egg and parmesan in separate shallow dishes.
Slice the courgettes into finger sized pieces. Dip each piece into the egg, then into the grated parmesan and place on a baking tray. Once you've put all your courgette fries on a baking tray, sprinkle on any parmesan that might be left over. Sprinkle with a pinch of salt and pepper and place in the oven. Cook for 15 minutes, then turn the oven down to 170c/325f. Turn the fries over using a spatula (a few may stick, so a metal spatula is best to prevent breaking the fries) and cook for a further 8-12 minutes until the courgette fries are a deep golden brown.
Whilst your courgette fries are in the oven, cook your eggs.
Place both eggs into a medium sized sauce pan and fill it with cold water until the water just covers them. Heat on a high heat until the water comes to a rolling boil. Once boiling, turn the heat off and set your timer for 3 minutes. After 3 minutes, take out of the water and serve immediately with the courgette fries.