Preheat a 5- or 6-quart air fryer to 350°F for 3 minutes.
In a large bowl, combine ground pork, bread crumbs, beaten egg, crushed garlic, chopped dill, caraway seeds, caramelized onion, and malt vinegar.
Season the mixture well.
Cut the puff pastry sheets in half.
Spoon or pipe equal amounts of the pork mixture lengthways along the center of each pastry piece.
Roll the pastry over to enclose the filling.
Cut each roll into four pieces.
Place the rolls, seam-side down, on a parchment paper-lined tray.
In a small bowl, combine egg yolks and 1 teaspoon of water.
Brush the pastry with the egg wash and sprinkle with extra caraway seeds.
Taking care, pull out the air fryer pan and basket, then spray the basket with cooking spray.
Place 8 sausage rolls in the basket.
Slide the pan and basket back into the appliance.
Keep the temperature set at 350°F and set the timer for 15 minutes.
Cook until the sausage rolls are puffed and cooked through.
Transfer them to a wire rack.
Repeat the cooking two more times with the remaining sausage rolls.
Serve the hot sausage rolls with barbecue sauce.
Notes / Tips / Wine Advice:
You can make caramelized onions in your slow cooker and freeze them or keep them in the refrigerator for up to 5 days. To make caramelized onions, put 3 pounds of sliced onions in a 5-quart slow cooker. Add 1/2 cup of unsalted butter and 1/2 teaspoon salt. Stir the mixture and cook on high for 8 hours.
Rolls can be prepared to the end of step 3 a day ahead; refrigerate until ready to bake. They can also be frozen in an airtight container for up to 3 months.