For the scones: Line a baking sheet with parchment paper.
In a large bowl, whisk together the flour, sugar, baking powder, salt and 1 teaspoon of the cinnamon. Using a pastry cutter or a fork, cut in the butter until the mixture is crumbly. Fold in the blackberries and peaches. In a measuring cup, whisk together 1/2 cup of the buttermilk and the eggs and add to the flour mixture, stirring gently to combine.
Turn the dough out onto a work surface dusted with flour and press into a 9-by-6-inch rectangle. Use a bench scraper or knife to make the sides even, then cut the dough into 6 equal pieces (3-inch squares). Cut each square on the diagonal into triangles and transfer the scones to the prepared baking sheet.
In a small bowl, stir together the remaining 1/2 teaspoon cinnamon with the coarse sugar. Brush the top of each scone with the remaining 2 tablespoons buttermilk and sprinkle with the cinnamon-sugar mixture. Place the baking sheet with the scones in the freezer for 20 minutes.
Preheat the oven to 425 degrees F.
For the glaze: Meanwhile, put the blackberries, granulated sugar and 1 tablespoon of the lemon juice in a small saucepan over medium heat. Bring to a simmer and cook until blackberries are softened and the sugar is dissolved, 5 to 8 minutes, mashing as needed to break up the berries.
Strain through a fine-mesh strainer into a medium bowl, pressing with a rubber spatula to release all the juice. Let cool slightly, then whisk in the confectioners' sugar and 2 tablespoons of the lemon juice until smooth. If necessary, add extra lemon juice a teaspoon at a time to adjust the consistency of the glaze.
Bake the chilled scones until golden, 18 to 20 minutes. Let cool slightly on the baking sheet, then drizzle with the glaze.