Prepare the filling: Place potato cubes in a medium saucepan, cover with water and season with 1 teaspoon salt. Bring to a boil, reduce to a simmer, and cook until cubes are completely tender, about 10 minutes. Drain and transfer to a medium bowl. Set aside.
Meanwhile, using a flat palm, press the cut side of the tomato against the large holes of a box grater and grate the tomato until you’re left with just the skin. (The skin should protect your hand from the grater.) Discard the skin, then transfer the tomato and any juices to a small bowl. Set aside.
In a 10- to 12-inch skillet over medium-high, heat vegetable oil until shimmering. Add beef or pork, and cook, breaking it up with a wooden spoon until the pieces are quite small and completely cooked through, about 5 minutes.
Add onion, scallion, paprika, bouillon, and a few grinds of black pepper, and cook, stirring, until the onion is softened but not browned, about 1 minute. Immediately add the grated tomato and its juices, and cook, stirring, until the mixture is quite dry, about 5 minutes.
Transfer meat mixture to the bowl with potatoes and stir together with a fork, letting the potato get roughly mashed up. (The filling should form cohesive balls when gently packed.) Taste and adjust seasoning with more salt, bouillon and pepper to taste. Allow to cool completely.
To make the masa: Combine masarepa, salt and 2 cups tepid water in a large bowl. Stir with your hands and massage into a smooth dough.
Take a small ball in your hand and press it flat. If the dough cracks, add more water, 1 tablespoon at a time, until it is smoother and workable. (Small cracks around the edges are OK.)
Cover masa and set aside to rest for at least 15 minutes before using.
Assemble the empanadas: Divide masa into 24 golf-ball-size balls. Working one ball at a time, place inside a plastic zip-top bag and press down with the bottom of a skillet to form a circle about 3 inches in diameter. Carefully remove from the bag and transfer to a clean cutting board or tray. Repeat with remaining dough balls, laying them out side by side. Cover the disks with a clean, lightly damp kitchen towel.
Working one empanada at a time, place about 2 teaspoons of meat mixture in the center of the disk. Lift the front and back edges and seal them at the top by gently pushing them together. Seal the empanada, working from the center outward until both ends are completely sealed. Repeat with remaining empanadas.
In a large Dutch oven or wok over high, heat 1 quart oil until it registers 375 degrees on an instant-read or frying thermometer. Carefully drop empanadas into the hot oil one at a time. You should be able to cook about six empanadas per batch. Cook empanadas, adjusting heat to maintain an oil temperature of 325 to 375 degrees, until empanadas are crisp and golden on one side, about 3 minutes. Flip empanadas over and cook another 2-3 minutes on the second side. Using a slotted spoon or tongs, transfer empanadas to a wire rack placed over a sheet pan and immediately season with salt if desired.
To make the aji: Finely mince the cilantro, onion, scallions and chile by hand, or by roughly chopping, then pulsing in a food processor. Transfer to a medium bowl.
Working directly in the medium bowl and using a flat palm to press the cut side of the tomato against the large holes of a box grater, grate the tomato until you’re left with just the skin. (The skin should protect your hand from the grater.) Discard the skin.
Add 1 teaspoon kosher salt and a few tablespoons of water. Squeeze with lime juice to taste. Stir to combine. The ají should be quite thin in texture. Add more water as necessary. Taste and season with more salt, if desired.
If a creamy aji is desired, mix ¼ cup aji with 2 TBS sour cream.