Heat the olive oil in a large skillet over medium heat. Add the onion and cook, stirring occasionally, until softened, about 3 minutes.
Add the ground beef and cook, breaking it into small crumbles, until no pink remains, about 5 to 6 minutes. Carefully drain off excess fat if needed.
While the beef cooks, combine the potato, rutabaga, carrot, and water in a microwave-safe bowl. Cover loosely and microwave on high for 4 to 5 minutes, stirring once, until the vegetables are just beginning to soften. Drain any water.
Stir the partially cooked vegetables, salt, pepper, thyme, and Worcestershire sauce into the beef mixture. Cook for 2 minutes, then taste and adjust seasoning if needed. Transfer the filling to a bowl or rimmed pan and let it cool for 10 minutes.
Heat the oven to 400°F (200°C). Line a large baking sheet with parchment paper. Unroll the pie crusts and keep them cold until needed.
Place one quarter of the cooled filling on one half of each pie crust round, leaving a 3/4-inch (2 cm) border. Brush the border lightly with water. Fold the empty half of the dough over the filling to form a half-moon, then press the edges closed with a fork or crimp them with your fingers.
Cut two small steam vents in the top of each pasty. Arrange the pasties on the prepared baking sheet and brush the tops with the egg wash.
Bake until deeply golden brown and crisp, 22 to 26 minutes. Rest for 5 to 10 minutes before serving, as the filling will be very hot.