In a large bowl, whisk together the flour and salt. Cut in the cold butter using a pastry cutter or fingers until mixture resembles coarse crumbs.
Gradually add ice water, one tablespoon at a time, mixing until dough comes together. Form into a disk, wrap in plastic, and refrigerate for at least 30 minutes.
Heat olive oil or butter in a small skillet over medium heat. Beat 3 eggs with salt, pepper, and chopped chives, then cook gently, stirring occasionally, until softly scrambled. Remove from heat and let cool slightly.
Preheat oven to 375°F (190°C) and line a baking sheet with parchment paper.
Roll out dough on a lightly floured surface to about 1/8 inch thick. Cut into rectangles roughly 4×3 inches, about 6-8 pieces.
Spoon scrambled eggs onto half the rectangles, then sprinkle shredded cheddar cheese on top, leaving a small border.
Brush edges with beaten egg wash, place a plain rectangle over each filled one, and press edges firmly with a fork to seal.
Brush tops with remaining egg wash and bake for 20-25 minutes until golden brown and puffed.
Let cool slightly before serving.