In a small frying pan with 1 tbsp olive oil, cook your burgers lightly, as if you were going to eat them medium-rare, pink in the middle. They will have lots of time in the oven to finish cooking.
Divide the burgers in half lengthways (if they are at least 2cm thick) and then into small circles (about 4cm in diametre) using a cookie cutter.
Preheat the oven to 200 degrees Celsius, 400 Fahrenheit, Gas Mark 6. In a processor, pulse the mushrooms to a small crumb (don’t overdo it or you will end up with a mush).
Place a large frying pan on medium heat and add 2 tbsp olive oil, the butter and the thyme sprigs. When the butter has melted add the mushroom crumb. Cook for 10 minutes until the mixture is soft.
Deglaze the frying pan with the white Port (you can use white wine) and let it cook for another 10 minutes in order to cook the alcohol.
After 10 minutes, the duxelle should be ready. You are looking for a mixture that is sticky enough to hold it’s shape. Taste to adjust seasoning and add some freshly grated black pepper, mixing well. Remove the stalks of thyme and let it cool.
Lightly dust your surface with flour and roll out the puff pastry until it is very thin (about 2mm thick). Cut it into squares of about 15cm by 15cm.
Lay 3 slices of prosciutto over the pastry, overlaying them slightly. Cut the excess prosciutto if it overhangs past the pastry square.
On top of the prosciutto, place 1 heaped tsp of mushroom duxelle in the middle and place the burger disc on top. Follow with a tbsp of grated cheddar.
In a bowl beat one egg yolk with 1 tbsp of water to make an egg wash. Fold the corners of the pastry square so they join in the middle and brush with a little egg wash to help them stick together.
Lay a sheet of baking paper on a baking tray and place your parcels on top. Brush them all with the egg wash before putting them in the oven.
Cook in the pre-heated oven for about 20 minutes or until golden brown. Remove from the oven and place in a rack to cool.