Place potatoes in a large pot and cover with water; bring to a boil. Reduce heat to medium-low and simmer until barely tender, about 8 minutes. Drain. Once cool enough to handle, peel and cut into cubes. Set aside.
Meanwhile, melt butter in a large skillet and cook onion and green onions until softened, about 5 minutes. Add bell pepper and cook for 3 minutes. Add ground beef; cook and stir until beef is browned, about 5 minutes. Season with paprika, cayenne pepper, and salt.
Remove beef mixture from heat; add potatoes and olives. Stir in chopped hard-cooked eggs; set filling aside to cool, about 20 minutes.
Preheat the oven to 425 degrees F (220 degrees C). Line 2 baking sheets with aluminum foil and lightly grease with olive oil.
Place an empanada pastry round on a flat surface and lightly wet the edges with warm water. Put a large tablespoon of cooled filling in the center and fold dough over. Seal edges by pressing with a fork. Repeat with remaining rounds and filling. Transfer empanadas to the prepared baking sheets and brush with beaten egg.
Bake in the preheated oven until golden brown, 15 to 20 minutes.