Heat oil in a skillet and saute the carrots, onion, potatoes, and garlic until al dente. Add ground beef and brown. Drain fat from pan and season with salt and pepper and jerk seasoning. Stir in tomato paste to bind mixture, and let cool.
Preheat oven to 375 degrees F.
Roll out puff pastry dough and, using a circle cutter, cut out 16 to 24 (3-inch) diameter dough circles and place on a lightly greased baking sheet. Place a spoonful of meat mixture on 1 side of each pastry circle. Fold each into a half-circle and seal edges by pinching the dough together. Brush surface of pastry with egg wash. Bake until golden brown, about 6 to 8 minutes.