In a large bowl, mix together flour and salt. Add the olive oil by the tablespoonful, cutting it into the flour until the flour appears pebbly.
In a cup, mix together 4 tablespoons of ice water with apple cider vinegar. Slowly add 2 tablespoons of the water and vinegar mixture into the flour and stir into the dough, adding more water a tablespoon at a time until it holds together to form a soft ball. Be careful not to over-knead the dough.
Roll the dough into a ball, then press it into a disk about an inch thick. Place the disk between two 14 inch long pieces of waxed paper. Use a rolling pin, roll the disk into a circle about 1/4 inch thick. Refrigerate the dough wrapped in waxed paper for at least 30 minutes, or until ready to use.
While the dough is refrigerating, toss the strawberries, raspberries, and basil together in a bowl. You may add a sprinkle of sugar if you like, depending on how sweet you want it (I found it wasn’t necessary). Once the dough is chilled, spread the mixture in the center of the dough, leaving about 1 inch on every side. Fold the dough toward the center, pressing firmly to seal. Sprinkle a light coating of granulated sugar over the crust and press down gently. Bake for 20 minutes, reduce the heat to 350*, then bake for another 30-35 minutes, until the filling is bubbling and the crust is golden brown. Cool completely and serve room temperature.