Preheat the oven to 350°F (177°C).
Spray an 8-by-8-by-2-inch (20-by-20-by-5-cm) cake pan with non-stick cooking spray or coat with 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. Whisk in the buttermilk.
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. Don't 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, rotating the pan after 20 minutes, until a tester inserted comes out with a few moist crumbs, 40 to 50 minutes.
Let the cake cool to room temperature before dusting with confectioners’ sugar and serving right from the pan. Add a dollop of freshly whipped cream or a scoop of vanilla ice cream, if desired. (The cake will keep wrapped in plastic wrap for three days on the counter - and some claim it gets better with age.)