Combine water and salt in a small saucepan over medium heat and warm through, stirring until salt is dissolved. Remove from heat and let cool for 2 to 3 minutes.
Combine sifted flour and melted margarine in the bowl of a food processor; pulse until crumbly. Add salt water gradually to the food processor and pulse until a soft dough ball forms that easily separates from the edge of the bowl. Add more water, 1 teaspoon at a time, only if needed. Press dough into a ball and wrap tightly with plastic wrap. Refrigerate for at least 30 minutes.
Preheat the oven to 425 degrees F (220 degrees C). Cover a baking sheet with aluminum foil and grease with vegetable oil. Set aside.
Bring a saucepan of water to a boil. Add cubed potatoes and cook until soft, 8 to 10 minutes.
Meanwhile, melt butter in a skillet over medium heat. Cook and stir onions and green onions until soft and translucent, about 5 minutes. Add red peppers and cook for 5 more minutes. Add ground beef and stir with a wooden spoon until fully browned, about 5 minutes. Season with salt, cayenne pepper, cumin, and paprika. Remove from heat.
Drain potatoes and add to filling. Toss in chopped eggs and let filling cool down for a few minutes.
Remove pastry from the fridge and knead gently on a floured surface. Divide dough in half and roll out each piece to 1/8-inch thickness.
Cut out 3-inch circles with a pastry cutter or sharp knife and wet the edges lightly with water. Add 1 large tablespoon of filling in the center of each disc. Fold pastry over, press edges together, and seal with a fork. Transfer empanadas to the prepared baking sheet.
Bake in the preheated oven until golden brown, 10 to 15 minutes. Remove from the oven and allow to cool just a little before serving.