In a large bowl, stir together the scone mix, cheese, and basil.
Add the cold water and stir with a wooden spoon until all the dry ingredients are moistened and have loosely come together to form a shaggy dough ball. You'll know the dough ball is ready when it's roughly holding together; if the dough is smooth or elastic then it's been overworked.
Transfer the dough ball to a floured surface and knead it 4-5 times to bring all the ingredients together. Then divide the dough ball into 4 equal-sized pieces.
Shape and flatten each piece into a circle approximately 1/2-inch thick and place on a parchment paper-lined baking sheet.
Spread 2 tablespoons of the relish on 2 of the circles, leaving a slight border around the edge. Then top each with one of the remaining circles, gently pressing the 2 halves together to seal.
Transfer the baking sheet to your freezer for 30 minutes to chill the scones.
Meanwhile, pre-heat your oven to 425° F.
In a small bowl, whisk together the egg and heavy cream. Set aside.
Remove the baking sheet from the freezer and brush the tops of each of the scones with some of the egg wash. Then transfer the baking sheet to your pre-heated oven and bake the scones for 16-18 minutes, until the scones are golden brown on top.
Let the scones rest on their baking sheet for 15 minutes. Then using a sharp knife, cut each circle into 4 wedges. Serve.