For the filling: Heat the olive oil in a large saute pan over medium heat. Saute the onion until translucent, about 2 minutes. Add the potatoes, garlic and tomatoes and cook for 3 minutes. Add the paprika and oregano and continue to cook for 2 to 3 minutes. Add the chicken broth and cook until some of the liquid has been absorbed and the potatoes are tender, 8 to 10 minutes. Reduce the heat to low and add the peas, chicken, hard-boiled eggs and salt and pepper to taste and cook until heated through, about 3 minutes. Set aside to cool.
For the dough: Put the butter, sugar, salt, cayenne and 2 cups water in a medium saucepan and place over medium heat. Cook until the butter melts and the water is warm, 4 to 5 minutes.
Put the flour in a food processor and slowly pulse in the warm butter mixture until a dough forms. Place the dough on a lightly floured work surface and form into a ball. Wrap in plastic and let rest in the refrigerator for 30 minutes.
Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper.
For the egg wash: Whisk together the milk and egg in a small bowl.
Portion the dough into balls a little bit larger than a golf ball and flatten them into disks. On a floured work surface, roll out into rounds 5 to 6 inches in diameter. Place 2 heaping tablespoons of the filling into the center of each dough round. Brush the sides of the dough with the egg wash. Fold one side of the dough over the other, pressing against the filling to remove any air pockets. Crimp the edges of the empanadas with a fork to seal them. Brush the outsides of the empanadas with the egg wash and place on the prepared baking sheets.
Bake until golden brown, 25 to 30 minutes. Let cool for 10 minutes.