Cut 8 slices of 1 cm thick from the loaf and brush 4 of them with oil on one side and the other 4 on both sides. Wash and dry the blackberries well.
Heat 2 tablespoons of oil in a non-stick pan and sauté the blackberries for a couple of minutes, crushing them with a wooden spoon until they are slightly wilted (watch out for splashes!!!).
Wash the basil, dry it and cut it into strips. Remove the rind from the Fontina and cut it into 8 slices. Place 2 slices of Fontina on the ungreased part of the 4 slices of bread.
Heat a large non-stick pan with a little oil and brown the oiled bread slices on both sides until golden brown. Remove them from the pan and keep them warm.
Add a little oil to the pan and place the bread slices with the oiled side facing the pan. Lower the heat to minimum, cover the pan and let the slices brown and the Fontina slowly melt.
When the cheese is almost completely melted, cover it with a spoonful of blackberries and some basil. Close the sandwiches with the golden slices, and let everything warm up for a few more moments. Remove the sandwiches from the pan and serve hot.