In a medium saucepan, heat oil and sauté onions for about a minute. Then, add garlic, thyme, and cayenne pepper. Stir for about another minute or two for the flavors to bloom (careful not to let it burn).
Place sausage or ground beef, parsley, salt, breadcrumbs, egg, and smoked paprika into a medium bowl, followed by the sautéed ingredients. Mix until thoroughly combined. Divide into 2 or 4 portions, depending on the size of your puff pastry sheets.
Unroll the thawed puff pastry onto a lightly floured board. Spoon one part of the sausage mixture and spread along the length of each strip, leaving about ¼-½ inch for the edge. Tightly roll the pastry around the sausage mixture and brush the edges with beaten egg to secure. Repeat the process with the remaining sheet(s). Let it rest in the fridge for about 10 minutes to make cutting easier.
Use a sharp knife to cut each roll into about 8 or more pieces. I cut the ones in the pictures into about 10 pieces each for 20 pieces in total, while in the video, I cut them into 6 pieces each, totaling 12 pieces.
Place the rolls on a baking pan or sheet if saving for later. Freeze, and when frozen, remove and place in freezer ziplock bags. They last about a month. You can bake them frozen but add 7-10 minutes more baking time.
Brush sausage rolls with the remaining egg wash. Place in a 400℉ (205℃) oven and bake for 25-35 minutes until the pastry puffs and the meat cooks through. Remove and serve hot, warm, or at room temperature. Pair with ketchup if desired.