Step 1

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.

Step 2

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.

Step 3

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

Step 4

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.

Step 5

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

Step 6

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

Step 7

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.

Step 8

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.

Step 9

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.

Step 10

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

Step 11

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.