To make the sauce, put all the ingredients into a heavy-bottomed saucepan, place it over a medium–high heat and bring it to the boil.
Lower the heat and simmer for about 30 minutes, until the cranberries have broken down a little and the liquid has thickened.
Transfer to a clean container or sealable jar and leave to cool for about 1 hour.
Season the turkey, then mould it into patties a little larger than the buns. Using a 3cm circular cutter, cut a hole in the centre of each patty and pack in the sausage meat, pressing any excess turkey around the edges.
Trim halve and finely shred the sprouts, then combine in a bowl with enough mayonnaise to bind them, season to taste, then set aside.
Preheat the oven to 180ºC/gas 4.
Brush a griddle pan with oil, place it over a medium heat and cook the burgers for 4 to 5 minutes on each side, until the juices run clear, then place in the oven for 5 minutes.
Cut the buns in half and warm them under the grill.
To assemble the burgers, spread the remaining mayonnaise on the bottom of each bun, sit a burger on top, spoon on a dollop of cranberry sauce and finish with the sprouts.