Oil 12 muffin cups or line with paper muffin liners.
In a large mixing bowl with an electric hand-held mixer on medium setting, blend Olivia's Lemon EVOO and sugar; add in eggs (one at a time) beating after each addition. Blend in Olivia's Blackberry balsamic.
In a separate bowl, stir together the flour, baking powder, baking soda, and salt.
With a wooden spoon, stir flour mixture into EVOO/sugar mixture. Gradually stir in buttermilk until dry ingredients are just moistened; do not over-mix.
Spoon batter into prepared muffin pan, filling each muffin cup about 2/3 full. Evenly spoon sliced blackberries onto the center of each muffin and sprinkle each with a little granulated sugar.
Bake blackberry muffins at 350°F for 18 to 20 minutes. Cool in pan on rack for 5 minutes. Gently turn blackberry muffins onto the rack. Serve muffins warm.