Combine the flour, sugar, yeast and a large pinch of salt in a mixer bowl. Lightly beat the 2 whole eggs and add the milk. On a low speed using the dough hook attachment, slowly add the eggs and milk to the dry ingredients until completely incorporated. On a medium speed, add the butter cube by cube and continue to mix for 4 minutes, occasionally scraping the side of the bowl, until the butter is totally combined and the dough is glossy and stretchy. It will be very soft but shouldn’t be sticky. Grease a large bowl and scrape in the dough, then cover with clingfilm and leave in a draft-free place for 1 hour until doubled in size.
Place the flour, butter and a pinch of salt in a food processor and pulse briefly until it resembles breadcrumbs. Add the egg along with 1 tbsp ice-cold water and pulse briefly again until just starting to come together. Tip the dough on to a floured surface and knead until it comes together. Flatten into a disk and wrap in cling film, then chill for 30 minutes.
Heat the oil in a large pan over a medium heat. Add the beef and cook, breaking up with a spoon, for 5 minutes until no longer pink. Transfer to a medium bowl with a slotted spoon, leaving the fat in the pan.
Add the onion and garlic and cook for 6-8 minutes, scrapping the bottom of the pan, until golden. Add the dried herbs and smoked paprika and cook for 1 minute. Add the stock, tomato purée, tomato ketchup, Worcestershire sauce and beef along with any accumulated juices to the pan. Bring to a gentle boil, then simmer uncovered for 15 minutes until thick and glossy. Season well and set aside.
Punch down the dough and tip on to your work surface. Knead for a minute, then bring together into a ball. Lightly grease a loose-bottomed 18cm cake tin with 3cm sides and add the dough into the middle. Cover loosely with a lightly greased pieced of cling film and leave somewhere warm for an hour to double in size.
Remove the pastry from the fridge. Roll out the pastry into a 5mm round, then carefully use it to line a loose-bottomed 18cm cake tin with 3cm sides. Gently press into the tin, then trim off the excess pastry. Prick the base with a fork and transfer to a fridge to chill for 30 minutes.
Preheat your oven to 200°C/fan 180°C/gas mark 6. Add a deep baking tray to the bottom of your oven and leave for 10 minutes to get hot. Keeping the brioche dough in the tin, carefully take off the cling film. Combine the separated egg yolk with 1 tsp water and brush over the top to glaze. Sprinkle over the sesame seeds, then transfer to the oven on a middle shelf. Add some cold water to the baking tray to create steam and close the door. Bake for 25 minutes or until deeply golden on top. Leave to cool.
Line the pastry with a circle of baking paper on the bottom and add baking beans. Transfer to the oven and bake for 15 minutes. Remove from the oven and take out the baking beans and baking paper. Put the pastry back in the oven and bake for a further 10 minutes or until lightly golden all over. Set aside to cool slightly.
Gently reheat the mince, then pour it into the pasty case. Sprinkle over the cheese. Use a bread knife to slice off some of the bottom of the brioche bun – it should look like a very large hamburger bun. Pop it on top of the pie, then place it back in the oven for 10-12 minutes so that everything has heated through and the cheese melts. Slice to serve.