Ham, Cheese, and Onion Empanadas

Mix butter, vinegar, salt, and 2 cups lukewarm water in a large bowl until well combined. Gradually add 6 cups flour, mixing with a wooden spoon or your hands until a shaggy dough forms.

Transfer dough to a lightly floured surface and knead until mostly smooth and no dry spots remain, about 2 minutes. Form into a disk, wrap in plastic, and chill until cold, at least 2 hours.

Do Ahead: Dough can be made 2 days ahead. Keep chilled.

Heat olive oil in a large skillet over medium-high until shimmering. Cook onions, stirring occasionally, until golden and softened, 5–7 minutes. Season with salt and pepper; mix in oregano. Transfer to a large bowl and chill until cool enough to handle, about 20 minutes.

Add ham to chilled onions and toss to combine. Taste and season with salt and pepper.

Toss mozzarella and cornstarch in a medium bowl to coat (this will prevent the mozzarella from liquefying as the empanadas cook).

Divide dough into 6 pieces. Cover all but 1 piece with plastic wrap and keep chilled. Roll out dough to a rectangle about 15x10" and 1/16" thick. Using 4½" cutter, punch out 6 rounds. Roll out scraps to 1/16" thick to punch out another round or two.

Place about 1 Tbsp. ham mixture in the center of dough circle. Top with 1 or 2 pieces of cheese, breaking cheese up if needed. Using your fingers, brush water halfway around edge of each round. Fold dry side up and over filling to create a semicircle. Pinch edges to seal; crimp with your fingers. Transfer to a parchment–lined rimmed baking sheet. Chill while you repeat with remaining dough and filling. Chill empanadas at least 20 minutes before frying.

Pour vegetable oil into a large heavy pot or Dutch oven fitted with deep-fry thermometer to a depth of 3". Heat over medium-high until thermometer registers 350°. Working in batches of about 5 at a time and adjusting heat to maintain oil temperature, fry empanadas, turning often, until deep golden brown, about 4 minutes total.

Transfer to a wire rack. Let cool slightly before serving.

Do Ahead: Onion mixture (without ham) can be cooked 2 days ahead. Transfer to an airtight container and keep chilled. Empanadas can be filled 2 days ahead. Cover and keep chilled.