Prepare the dough: Place the flour, salt, and sugar into a bowl of an stand mixer fitted with the paddle attachment. Beat in the butter at low speed until mixture is well combined yet lumpy. Beat in the eggs, and then the water, until obtaining a homogeneous dough. You might need to stop the mixer a few times to scrape together dough from both the bottom and sides of the bowl.
Place the dough onto a floured surface, form a ball, then press into a flat disk. Pressing the dough into a disc allows it to chill faster. This will also make the dough easier to roll out, and if you freeze it, it will thaw more quickly. Wrap in plastic wrap and refrigerate for a bout 1 hour.
Meanwhile, prepare the pie filling. In a large non-stick saucepan, heat the oil over medium heat. Add the onion and let cook, stirring occasionally, for about 2-3 minutes or until softened. Add the garlic and cook, stirring occasionally, for about 1 minute or until fragrant. Add the chicken and sautée for about 1-2 minutes. Stir in the tomatoes, peas, corn, and green onions. Turn off the heat and set aside.
In a medium to large pot, combine well the chicken stock, tomato paste, and wine. Bring mixture to a boil over high heat and reduce the heat to low.
In a large saucepan over low heat, melt the butter and add the flour all at once, stirring to make a roux (approximately 2 minutes). Increase the heat to medium, add the stock, whisking constantly, until it thickens. Season with salt and pepper to taste. Combine mixture with the chicken mixture and set aside.
Remove the dough from the refrigerator at least 15-20 minutes before rolling.
To assemble the chicken pot pies, cut the dough into four pieces. On a lightly floured surface or on a surface covered with a plastic sheet, roll the dough into circles, a bit larger than the pie pan. Always work the dough from the center towards the edges. Unroll the dough onto a 9-inch pie pan and up the sides, leaving some extra dough hanging on the edges. Spread half of the filling inside the pan. Using a pastry brush or your fingers, lightly moisten the edges of the bottom crust with water. Fold a bit of extra top dough over the rim of the pan. Press the top crust against the bottom crust to seal and make a pattern on the edges with your fingers, then trim the overhang from both crusts. Using a paring knife, cut an X or slits in the center of the crust to vent while baking.
Repeat step 7 with the second pie. Place the empadao in the refrigerator for 30-60 minutes before baking. At this point, you can either bake the pie, or cover it with plastic wrap and store it in the refrigerator for up to 2 days or in the freezer for up to 3 months.
Preheat the oven to 350* F. Place the chicken pot pie on a baking sheet pan and bake, on the center rack of the oven, for about 30 minutes. Remove from the oven, brush the top crust with the egg wash and return to the oven for additional 10-15 minutes or until the crust is golden brown.
Transfer the chicken pot pie to a rack and let it rest for 15-20 minutes before serving. Enjoy the pie served warm, accompanied by a fresh salad or white rice.