Drain two cans of peaches and discard the syrup. Add the drained peaches and the third can with its syrup to a large pot along with 1/2 cup salted butter over medium heat. Stir until the butter melts.

Stir in 1 cup sugar, cinnamon, nutmeg, and vanilla extract. Bring the mixture to a boil. Remove 3 tablespoons of the peach syrup, whisk in 3 teaspoons flour to make a slurry, then stir it back into the pot. Cook for 30 to 40 minutes until the syrup thickens. Adjust seasonings and let cool.

Make the pie crust: combine 2 1/2 cups flour, 2 teaspoons sugar, and salt in a bowl. Cut the cold butter cubes into the flour with a pastry cutter until pea-sized crumbs form. Gradually add cold water until the dough comes together. Divide in half, shape into discs, wrap in plastic, and refrigerate for 35 to 45 minutes.

Preheat the oven to 375°F. Roll out one dough disc to about 1/3-inch thickness and line a 9x13-inch baking dish, pressing the dough up the sides.

Spoon the cooled peach filling onto the crust and pour the remaining thickened syrup over the top. Roll out the second dough disc and place it on top. Crimp the edges to seal.

Brush the top crust with egg wash and sprinkle with cinnamon. Bake for 40 to 45 minutes until the crust is golden brown and the filling is bubbly. Cool before slicing and serving.