Preheat the oven to 350°F. Spray a light-colored, non-stick 9x13 metal baking pan with baking spray (flour should be one of the ingredients, which is what makes it extra effective).
In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.
2 ⅔ cups all-purpose flour, 2 ½ teaspoon baking powder, ½ teaspoon kosher salt
In the bowl of a stand mixer fitted with a paddle attachment (or using a hand mixer), beat together the butter and sugar on medium speed until pale and fluffy, 4-5 minutes. Scrape the bowl halfway through.
1 cup butter, 2 cups granulated sugar
Add the eggs one at a time, mixing on medium-low speed until just combined after each addition. Then add the vanilla and mix until just combined. Scrape the bowl well, making sure to get the bottom.
4 eggs, 2 teaspoon vanilla extract
Add half of the flour mixture and combine on low speed until mostly combined. Then add the sour cream and continue mixing on low until almost combined. Lastly, add the remaining flour mixture and finish mixing on low until it's almost fully combined, but a few white streaks still remain.
1 cup full-fat sour cream
Add the fresh berries to the bowl and use a rubber spatula to gently fold them into the batter until they are evenly distributed. Be careful not to overmix; the batter will be pretty thick.
3 cups fresh mixed berries (raspberries, blackberries, and blueberries)
Pour the batter into the prepared pan and spread it into an even layer, then sprinkle the coarse sugar evenly across the top. Bake for 40-45 minutes, or until it is golden brown and a toothpick comes out clean (tent it loosely with foil if it is getting too dark in the last 5-10 minutes). Cool the cake completely in the pan.
3 tablespoon coarse sugar or sanding sugar