Combine flour and 1/4 teaspoon salt in a bowl. Add the butter and blend together with your fingertips or a pastry blender (or pulse in a food processor), until most of mixture resembles coarse meal with some small (roughly pea-size) lumps. Drizzle evenly with 4 tablespoons water and gently stir with a fork (or pulse in food processor) until incorporated. (It may still appear crumbly. Press together with fingers; if it doesn't clump, add another tablespoon of water.) Shape into a disc and cover with plastic wrap. Chill for 30 minutes.
Set oven temperature to 425 degrees.
Heat oil in a large stainless steel skillet over medium high heat until shimmering. Add onion, peppers and garlic and cook, stirring, until soft, about 4 minutes. Add potato, corn and 1/2 cup water. Bring to a simmer and cover with a lid. Cook, stirring often until potatoes are just soft, about 8 minutes. Let cool. Stir in cheese.
Divide dough into 8 pieces. Working with 1 piece at a time (cover remaining pieces with a damp towel to prevent drying), roll each dough portion into a 5-inch circle. Spoon 3 tablespoons potato-corn filling into center of each circle. Moisten edges of dough with water; fold dough over filling. Press edges together with a fork or fingers to seal. Place empanadas on a large baking sheet lined with parchment. Combine egg white and 1 tablespoon water. Lightly coat tops of empanadas with egg mixture. Pierce top of dough with the tip of knife. Bake for 20 minutes or until lightly browned.