Cut the butter into 1/4-inch thick slices; combine in the bowl of a food processor with the flour and salt. Pulse until the mixture resembles coarse crumbs; drizzle in water with the machine running, 1 tablespoon at a time, until the dough resembles lumpy oatmeal. Gather the dough into a ball, cover with plastic wrap, and refrigerate for at least 30 minutes.
Preheat oven to 375 degrees F (190 degrees C). Line a baking sheet with parchment paper.
Place each chicken breast between two sheets of heavy plastic (resealable freezer bags work well) on a solid, level surface. Firmly pound the chicken breast with the smooth side of a meat mallet to a thickness of 1/2 inch. Spread each chicken breast with a third of the cheese and spinach, and fold each chicken breast in half. Sprinkle each chicken breast with 1 tablespoon of basil, 1 teaspoon of garlic powder, and a pinch of garlic salt.
On a well-floured work surface, roll the dough ball out into a rectangle about 1/8-inch thick and cut into 3 equal-size pieces. Wrap each folded chicken breast in pastry and pinch the edges together to enclose the chicken in the pastry dough. Place the wrapped chicken breasts onto the prepared baking sheet, brush each pastry package with beaten egg, and bake until the chicken is no longer pink inside and the crust is golden brown, 40 to 45 minutes.
