Add the flour, sugar and salt to a food processor and pulse just until combined. In a small bowl, whisk together the egg, vinegar and water. Add the cold butter pieces into the food processor and pulse until small coarse crumbs remain. Sprinkle the water/egg mixture over the flour and pulse again until the dough comes together.
Remove the dough with your hands and wrap it in plastic wrap. Stick it in the fridge for 30 minutes.
While the crust is in the fridge, make the fruit filling. If you’re doing red and blue separately, then you will want to use 2 saucepans, 1 for each fruit. In each saucepan, combine the fruit with 2 tablespoons of the sugar and a big pinch of salt and cinnamon. Cook, stirring often, until the berries soften and get syrupy, about 10 minutes. Let the mixture cool completely.
Preheat the oven to 375 degrees F. Line a baking sheet with parchment paper. Fill a small bowl with water.
Roll out the dough until it's about 1/4 inch in thickness. Use a 2-inch biscuit cutter (or something else round, like a jar) to cut rounds out of the dough. Make sure to cut an equal amount since you will press them together!
Place the rounds on the baking sheet. Fill every other round with a tablespoon of the strawberry or blueberry mixture. Dip a finger in the water and run it along the outside of the dough round that has the peach mixture. Place an empty round on top of the berry filled round. Press the outsides down with a fork to seal the rounds together. Repeat with the remaining rounds. (Note: If you have extra berry mixture (you most likely will!) save it for serving. I LOVE to warm it back up and serve over the pies and ice cream.)
Beat the egg and water together. Brush it on top of all the pies and sprinkle with coarse sugar. Cut an "x" in the top with a paring knife.
Bake the pies for 20 to 25 minutes, or until golden brown. Remove and let cool slightly. Serve with vanilla ice cream!