In a small bowl, add the soy sauce, ketchup, beef base and water. Mix well. Set to the side.
Smash the potatoes but not so much that it turns into mashed potatoes. Season generously with salt and pepper. Set to the side.
In a large skillet over medium heat add 1 tablespoon of vegetable oil. Add ground beef chopping the beef with spoon as it cooks. You don’t want big chunks of ground beef. Cook beef until it’s browned. To make the filling look more authentic you can put the cooked cooled beef in a food processor and process until the meat is finely minced. Drain most of the liquid and leave only 2-3 tablespoons of fat in the skillet. Remove the beef and set to the side. Add the peppers, green and red onions. Sauté until the onions have softened, about 2-3 minutes. Add the garlic. Cook until garlic is fragrant. Set stove to medium-high heat. Return beef to the skillet. Add the liquid beef base mixture, seasonings and panko breadcrumbs. Mix well. Let it cook for a few minutes. Set to the side to cool.
Add one tablespoon of the smashed potatoes to the center of the dough. Next, add 2-2.5 tablespoons of beef filling to the center of each circle of dough, leaving some room free around the edges. Brush a bit of egg wash around the edges of the dough.
Take one side of the dough and carefully fold it over to the other side. You may need to gently pat the beef filling down and move it off center some. Using the tinges of a fork press down on the dough to seal it. Brush the top and the edges of each patty with egg. Place patties on a baking sheet and freeze for about 45 minutes to an hour, or overnight. See notes below.
Preheat oven to 375 degrees Fahrenheit. Bake for 25-30 minutes or until crust is golden brown and the shell is hard and not soft. Serve once they’ve cooled off some.