Prep. Preheat the oven to 400 degrees Fahrenheit (204 degrees Celsius). To make cleanup easier, line a baking pan with aluminum foil. Sprinkle some flour on the pie plate and gently place the bottom crust, then carefully put the bottom crust into it, cutting off any excess.
Make the Filling. Combine the blackberries, raspberries, and sugar in a large saucepan over medium heat. Stir for around 5 minutes until the sugar dissolves and the berries release their juices. Once done, taste the mixture and add more sugar, if needed, one tablespoon at a time. Mix cornstarch and lemon juice in a separate bowl to make a smooth slurry. Gradually pour this mixture into the simmering berries while stirring continuously. Continue to cook and stir for 2-3 more minutes until the mixture thickens slightly.
Assemble the pie. After preparing the berry mixture, let it cool for 10 minutes so the bottom crust doesn't become soggy. Then, pour the slightly cooled berry filling into the pie crust that you have prepared. Cover the filling with the top crust and seal the edges by crimping with a fork or your fingers. Make small slits in the top crust to allow steam to escape. Finally, brush the top crust with milk and sprinkle coarse sugar for a shiny and sweet finish.
Bake. Put the pie on a baking sheet you've prepared earlier and bake it in an oven that you have preheated for 40-45 minutes. Make sure it turns golden brown and the filling bubbles through the slits. If the edges of the crust become brown too quickly, cover them with foil strips or a pie crust shield halfway through baking.
Chill and serve: After baking the pie, put it on a cooling rack and let it cool for about 2 hours at room temperature. Then transfer it to the fridge and let it chill throughly before serving. If you skip this step, it will be very runny.