In a large bowl, whisk the warm water, fine salt, and sugar until dissolved, 30–60 seconds.
Gradually sprinkle in the masarepa while stirring, then add 1 tbsp vegetable oil (for dough). Mix and knead until a smooth, soft dough forms that doesn’t crack when pressed, 2–3 minutes. Cover and rest to hydrate, 10 minutes.
While the dough rests, shred the queso blanco on the large holes of a grater, about 3 minutes.
Divide the dough into 10 equal balls (about 60 g each). Working one at a time, place a ball between two pieces of plastic (cut from a zip-top bag) and press or roll into a 5.5–6 inch disc, about 1/8 inch thick, 8–10 minutes total for all discs.
Spoon about 2–3 tbsp shredded cheese onto one half of each disc, leaving a 1/2 inch border. Fold the dough over to form a half-moon, press to expel air, and seal the edge firmly; trim with the rim of a bowl or crimp with fingers for a tight seal, 3–5 minutes.
Heat 6 cups vegetable oil (for frying) in a deep, heavy pot to 350°F/175°C over medium heat, 8–10 minutes. The oil is ready when a pinch of dough sizzles immediately.
Fry empanadas in batches without crowding, turning once, until deep golden and crisp and the cheese is melted, 4–6 minutes per batch. Maintain 330–350°F/165–175°C.
Drain on paper towels until the surface looks dry and crisp, 2 minutes. Serve hot with hot sauce and lime wedges.