In a large skillet over medium heat, add a drizzle of olive oil and brown the ground beef for about 5-7 minutes. Stir in the chopped yellow onion, minced garlic, fresh thyme, and sliced escallion, then season with paprika, ground allspice, salt, and pepper. Simmer for an additional 5 minutes until the mixture thickens; allow it to cool completely.

In a mixing bowl, whisk together the all-purpose flour, turmeric powder, and granulated sugar. Cut in the cold shortening and butter until the mixture resembles coarse crumbs. Gradually stir in ice-cold water, 1 tablespoon at a time, until the dough forms a ball. Wrap tightly in plastic wrap and chill in the refrigerator for 30-60 minutes.

Preheat the oven to 350°F (175°C). Roll out the chilled dough to about ⅛-inch thickness on a lightly floured surface. Use a round cutter to cut the dough into 5-6 inch circles. Place 2-3 tablespoons of the cooled meat filling on one side of each circle. Carefully fold over the dough and crimp the edges with a fork to seal securely.

Arrange the assembled Jamaican Beef Patties on a baking sheet lined with parchment paper. Brush the tops with an egg wash made from a beaten egg. Bake in the preheated oven for 30-32 minutes, or until they are beautifully golden brown and flaky. Allow to cool slightly before serving.