Preheat the oven to 350 degrees and spray an 8x8x2-inch pan with non-stick cooking spray or softened butter. Line with parchment paper.
In a large bowl, whisk together the oil, sugars, and vanilla. Add the egg, and then the yolk, whisking after each. Add the buttermilk and whisk a final time.
Sift the flour, baking powder, and salt onto a sheet of parchment, and using the parchment as a funnel, add the dry ingredients to the wet in three installments, folding with a flexible spatula after each, just to incorporate. Do not overmix.
Pour the batter into the prepared pan.
Sprinkle the top of the cake with the berries and then with the Turbinado sugar.
Transfer the cake to the oven and bake for 35 to 40 minutes, rotating the pan after 20 minutes. The cake is done when a tester inserted into the cake comes out with a few moist crumbs.
Bring to room temperature before dusting with confectioners’ sugar and serving along with a dollop of freshly whipped cream, or not, or a scoop of vanilla ice cream, or not. The cake will keep wrapped in plastic wrap for three days on the counter - and some claim it gets better with age.