Follow these detailed steps to craft your perfect mini veggie empanadas. The process is divided into three main stages: making the dough, preparing the filling, and the final assembly and baking.

Part 1: Preparing the Flaky Empanada Dough

1. Combine Dry Ingredients: In a large mixing bowl, whisk together the 3 cups of all-purpose flour and 1 teaspoon of salt.
2. Cut in the Butter: Add the cold, cubed butter to the flour mixture. Using a pastry cutter or your fingertips, work the butter into the flour until the mixture resembles coarse breadcrumbs with some pea-sized pieces of butter still visible. It’s crucial not to overmix; these butter pockets are the secret to flakiness.
3. Add Wet Ingredients: In a small bowl, whisk together the large egg, 1/2 cup of ice water, and 1 tablespoon of vinegar. Pour this liquid mixture over the flour and butter mixture.
4. Form the Dough: Use a fork or a spatula to gently mix until a shaggy dough begins to form. Do not overwork it. Turn the dough out onto a lightly floured surface and gently knead it just 2-3 times to bring it all together into a cohesive ball.
5. Chill the Dough: Divide the dough in half, flatten each half into a 1-inch thick disk, and wrap them tightly in plastic wrap. Refrigerate for at least 1 hour, or up to 2 days. This chilling period is essential for relaxing the gluten and firming up the butter, ensuring a tender and flaky crust.

Part 2: Crafting the Savory Vegetable Filling

1. Sauté the Aromatics: Heat the 2 tablespoons of olive oil in a large skillet or pan over medium heat. Add the finely diced onion and cook until it becomes soft and translucent, about 5-7 minutes. Add the minced garlic and cook for another minute until fragrant.
2. Cook the Vegetables: Add the finely diced carrots and potatoes to the skillet. Sauté for 5-6 minutes to give them a head start on cooking.
3. Add Spices and Broth: Stir in the smoked paprika, cumin, and dried oregano, cooking for about 30 seconds until the spices are fragrant. Pour in the 1/2 cup of vegetable broth to deglaze the pan, scraping up any browned bits from the bottom. Bring the mixture to a simmer, then reduce the heat to low, cover the skillet, and let it cook for 10-12 minutes, or until the potatoes and carrots are tender.
4. Incorporate Final Veggies: Stir in the frozen corn and frozen peas. Cook for another 3-4 minutes until they are heated through.
5. Season and Cool: Remove the skillet from the heat. Season generously with salt and black pepper to your taste. Transfer the filling to a bowl and let it cool completely to room temperature. You can speed this up by placing it in the refrigerator. This is a critical step! A warm filling will melt the butter in your dough, resulting in soggy, greasy empanadas.

Part 3: Assembling and Baking the Empanadas

1. Preheat and Prepare: Preheat your oven to 400°F (200°C). Line two large baking sheets with parchment paper.
2. Roll Out the Dough: Take one disk of chilled dough from the refrigerator. On a lightly floured surface, roll it out to about 1/8-inch thickness.
3. Cut the Discs: Using a 3-inch round cookie cutter (or the rim of a glass), cut out as many circles as you can. Gather the scraps, gently re-roll, and cut more circles until all the dough is used. Repeat with the second disk of dough.
4. Fill the Empanadas: Place about 1 to 1.5 teaspoons of the cooled vegetable filling in the center of each dough circle. Be careful not to overfill, as this can cause them to burst open while baking.
5. Fold and Seal: Gently fold one side of the dough circle over the filling to meet the other edge, creating a half-moon shape. Press the edges together firmly to seal.
6. Crimp the Edges: To ensure a tight seal, use the tines of a fork to press down along the curved edge. Alternatively, for a traditional look, you can create a decorative rope-like border known as a repulgue by folding small sections of the edge over themselves.
7. Arrange and Egg Wash: Place the assembled empanadas on the prepared baking sheets. In a small bowl, beat the remaining egg with 1 tablespoon of water to create an egg wash. Brush the tops of each empanada with the egg wash. This will give them a beautiful shine and a deep golden color.
8. Bake to Perfection: Bake for 20-25 minutes, or until the empanadas are puffed up and a deep golden brown.
9. Cool and Serve: Let the empanadas cool on the baking sheet for a few minutes before transferring them to a wire rack. They can be served warm or at room temperature.