To make the dough:
In a bowl, combine flour, salt, and butter. Using a pastry blender or fork, cut the butter into the mixture until it has a coarse, granular appearance. Make a well in the center of the mixture and add the egg. Mix well to incorporate the egg into the mixture, and add 1 to 2 tablespoons of ice water, adding just enough to make soft dough that is firm and minimally sticky. Knead the dough briefly and refrigerate for at least 1 hour.
To make the empanadas:
Divide the dough into 10 equal pieces and roll into balls. Lightly flour surface, roll out balls to make circles about 6 inches in diameter.
Place guava paste (about 1 ½ ounce) onto the center of each circle.
Using a pastry brush or finger, lightly wet the inside bottom edge of each circle and fold the top end over the filling.
Press the edges firmly together, flute them with the tines of a fork, and prick the tops. Trim the edges with a knife tip if necessary.
To fry: Heat oil in a large skillet until hot, and fry each turnover for approximately 1 minute on each side or until golden brown. Drain onto paper towels and serve warm.
To bake: Preheat the oven to 400°F. Lightly grease baking sheets. Brush the tops of each turnover with egg wash and place on baking sheets. Bake for 12 to 15 minutes, for until lightly browned.