
Gently warm the milk until it’s lukewarm.
Put the flour in the bowl of your stand mixer and add the yeast, sugar and salt (don’t place the yeast directly on top of the salt). Give it a quick mix with a fork.
Add the 2 egg yolks and lukewarm milk, then knead using the hook attachment on a medium speed until it forms a rough dough (about 5 minutes).
Next, with the mixer on, add the butter a little at a time. Once all the butter is incorporated, knead on a medium speed for 10 minutes. The dough might look quite wet and sticky, but it comes together into a smooth and elastic dough after 10 minutes of kneading.
Grease a mixing bowl with butter. Form the dough into a ball and place it in the mixing bowl. Cover with plastic wrap (cling film) and let it proof for 3 hours or until doubled in size.
Grease your springform cake tin with butter and set aside. Cut the mozzarella, ham and other cheeses you are using into small cubes; set aside.
Tip the dough from the bowl onto a clean work surface and cut it in half. Roll each half into logs around 12 inches in length. Cut each log into 8 equal pieces, giving you 16 pieces in total.
Working with one piece of dough at a time, flatten it with your hand into a rough circle and fill it with some ham and cheese (don’t overstuff). Bring the edges together and pinch it at the top to seal the filling inside.
Turn the dough over, so it’s seam side down on your worktop. Cup your hands around the dough and gently rotate it in a circular motion, keeping the edges tucked under to form a smooth ball, then place it in the cake tin.
Repeat this with the rest of the dough until you have 16 filled buns in your cake tin. Cover with plastic wrap (cling film) and leave to proof until doubled in size (around 2-3 hours).
Preheat your oven to 180℃/350℉ (fan or conventional).
Whisk the egg and milk together in a bowl, then lightly brush it over the brioche.
Bake in the oven for 30 minutes if using a fan oven or 35-40 minutes if using a conventional oven, until golden brown on top (see my notes on this). Let it rest for at least 10 minutes before serving.