Place the flour, butter and 1 tsp salt in a food processor and pulse until it resembles breadcrumbs. Add the egg along with 1 tbsp of water and pulse again once or twice, until just starting to come together. Tip the dough on to a floured surface and knead briefly until it comes together. Flatten into a disk and wrap in clingfilm, then chill for 30 minutes.
Preheat the oven to 190°C/fan 170°C/gas mark 5. Place the eggs in a medium saucepan and cover with cold water, then bring to a boil and simmer for 4 minutes. Drain and rinse under cold water until cool. Carefully peel and set aside.
Roll out the pastry into a 5mm-thick round, then carefully line a 21cm tart tin. Gently press into the tin, then trim off the excess pastry. Transfer to a fridge to chill and start on the filling. Remove the sausages from their skins and add to a medium bowl. Add two-thirds of the parsley and season. Mix well until combined.
Remove the tart case from the fridge and add just under a third of the sausage mixture, spreading across the bottom evenly using a rubber spatula or your hands. Dip them in water to avoid everything getting sticky. Create a circle in the sausage mixture by making a groove in the base of the pie, then place the eggs in the groove. Cover with the remaining sausage mixture scoop by scoop, making sure the eggs are fully covered.
Combine the breadcrumbs, remaining parsley, parmesan and olive oil in a bowl, then sprinkle over the top of the sausage mixture. Place in the oven and bake for 1 hour 10 minutes, covering with foil towards the end if it gets too dark. Leave to cool in the tin before serving.