Preheat oven to 425°F (220°C)
Drain two cans of peaches, reserving the syrup. Add a third can of peaches and drain as well.
In a large saucepan, combine the drained peaches, reserved syrup, sugar, cinnamon, nutmeg, and vanilla extract. Heat gently, stirring occasionally.
Mix the 3 teaspoons of all-purpose flour with 2 tablespoons of cold water to create a slurry. Add the slurry to the saucepan and stir until the filling thickens, about 5–7 minutes. Remove from heat.
In a large mixing bowl, combine the 2 ½ cups flour, 2 teaspoons granulated sugar, and 1 teaspoon salt for the crust. Add the cold, cubed butter and mix until the mixture resembles coarse sand.
Slowly add the cold water, a few tablespoons at a time, until the dough comes together. Knead gently and chill in the refrigerator for 30 minutes.
Roll out the dough on a floured surface and press it into the bottom of a 9-inch round or deep-dish baking dish. Pour the warm peach filling into the crust.
Roll out the second half of the dough and cut into strips or shapes of your choice to create a decorative lattice or top crust.
Brush the crust with the egg wash (1 egg mixed with 1 teaspoon water).
Sprinkle a little ground cinnamon over the cobbler before baking.
Place on a baking tray to catch any drips and bake for 55–65 minutes, or until the crust is golden brown and the filling is bubbly.
Let cool slightly before serving.