In a large bowl, mix the flour and salt. Add the cold, cubed butter. Use a pastry cutter or your fingers to cut the butter into the flour until the mixture resembles coarse crumbs.
Gradually add ice water, one tablespoon at a time, and mix until the dough just comes together. Divide the dough into two equal parts, form each into a disc, and wrap them in plastic wrap. Refrigerate for at least 1 hour.
Preheat the oven to 400°F (200°C). Cook the chicken breasts until fully cooked, then dice into bite-sized pieces.
In a large saucepan, melt the butter over medium heat. Add the chopped onions and minced garlic, cooking until softened. Stir in the flour, salt, black pepper, thyme, and rosemary until well combined.
Gradually whisk in the chicken broth and milk, stirring constantly to avoid lumps. Cook until the mixture thickens and becomes a smooth, creamy sauce.
Add the diced chicken and frozen mixed vegetables to the sauce, stirring until evenly coated. Simmer for 5-7 minutes until the vegetables are heated through.
Roll out one disc of pie dough on a floured surface. Place it into a pie dish, trimming any excess hanging over the edges. Pour the chicken and vegetable filling into the pie crust. Roll out the second disc of pie dough and place it over the filling. Trim the excess dough, then crimp the edges to seal the pie. Cut a few slits on the top to allow steam to escape during baking.
Place the assembled pies on a baking sheet to catch any spills. Bake in the preheated oven for 30-35 minutes or until the crust is golden brown and the filling is bubbly.
Allow the chicken pot pies to cool for a few minutes before serving. Enjoy the comforting warmth and savory flavors of this classic dish!