Make the dough. In a large bowl, whisk whole-wheat pastry flour, all-purpose flour, and salt. Cut in the cold butter with a pastry cutter or your fingertips until pea-sized crumbs form. Add ice water 1 tablespoon at a time, mixing just until the dough holds together when pinched.
Chill. Turn the dough onto a lightly floured surface, press into a disc, wrap, and chill for 45–60 minutes. This relaxes gluten and keeps the butter cold for flakiness.
Cook the filling. Heat olive oil in a skillet over medium. Add onion and bell pepper; cook 5–6 minutes until softened and lightly golden. Stir in carrot and garlic; cook 1–2 minutes until fragrant.
Add greens and corn. Toss in spinach and corn. Cook until spinach wilts and moisture mostly evaporates, about 2–3 minutes. Season with smoked paprika, cumin, crushed red pepper, salt, and black pepper. Remove from heat and cool completely. Stir in shredded cheese once cooled.
Preheat and prep. Heat oven to 400°F (200°C). Line two baking sheets with parchment.
Roll the dough. On a lightly floured surface, roll the chilled dough to about 1/8 inch thick. Use a 3.5–4 inch round cutter (or a glass) to cut circles. Gather scraps, chill 5 minutes if warm, and re-roll.
Fill and fold. Place about 1 tablespoon of filling in the center of each circle. Fold over into a half-moon, gently pressing out air. Crimp edges with a fork or pleat by hand for the classic empanada look.
Egg wash and top. Arrange on prepared sheets. Brush with beaten egg. Sprinkle optional toppings like sesame or everything seasoning for extra crunch and style points.
Bake. Bake 16–20 minutes until deep golden brown and crisp. Rotate pans halfway for even color. If they look pale, give them 2–3 more minutes. Golden or bust.
Rest and serve. Let cool 5 minutes so the cheese settles. Serve hot with your favorite dip: chimichurri, salsa verde, or a garlicky yogurt sauce.