Preheat the grill or griddle-pan to medium heat. Brush the cut side of the peaches lightly with canola oil. Place cut side down on the grill for 2 minutes then rotate 90 degrees and continue cooking until there are grill marks and the fruit starts to caramelize, about an additional 1 to 2 minutes.
Slice the peaches and place them and the blackberries in a large mixing bowl. In a separate bowl, combine the sugar, flour, cinnamon, and salt. Sift the dry ingredients over the fruit and gently fold them together with a spatula.
Let the fruit mixture sit until the fruit is macerated, about 30 minutes. Discard the natural juices that have gathered in the bowl. Add the filling to an 11-inch x 7-inch baking dish.
For the crust:
Preheat oven to 375ºF (190ºC).
In a medium bowl whisk together the flour, sugar, baking powder and salt. Using your hands or a pastry blender cut the butter in until the mixture forms coarse crumbs.
You should still be able to see pieces of butter in the mixture. Stir the eggs into the crust and drop by spoonfuls on top of the filling.
Place a baking sheet under the baking dish in the oven just in case the mixture spills over. Bake the cobbler for until the filling is bubbling and the crust is golden brown and a toothpick inserted into the center of the top comes out clean, about 1 hour.
To serve:
Ladle into bowls and top with salted caramel ice cream.