Preheat the oven to 350°F. Spray the bottom of a 9-inch cake pan with cooking spray and line with parchment paper (either cut to fit the bottom, or leaving some hanging over the edges for easy removal).
In a medium bowl, whisk the flour, 1 ¼ cup oats, baking powder, and salt together.
In a stand mixer fitted with the paddle (or in a large bowl with an electric hand mixer), beat the butter, brown sugar, ¼ cup granulated sugar, and vanilla together on medium-high, periodically scraping down the sides of the bowl to make sure everything incorporates, until the mixture is pale, light, fluffy, and creamy, about 5 minutes.
Add the eggs, one at a time, beating to blend after each addition. (This is a good time to scrape down the sides again.)
Reduce the mixer speed to low and carefully add half the oat mixture, followed by half the buttermilk. Repeat with the remaining flour mixture and buttermilk, beating just until no large lumps remain.
Using a spatula, gently fold in about ¾ the fruit and transfer the batter to the prepared cake pan. Scatter with the remaining fruit and sprinkle with remaining ¼ cup oats and 3 tablespoons granulated sugar. Bake until the cake is puffed, deeply golden brown, and pulling away at the sides, 65–75 minutes. (It should spring back slightly when pressed in the center and appear fully baked, especially in the center, where the fruit meets the cake.)
Let the cake cool completely (really, all the way) before slicing.
DO AHEAD: Cake keeps well for 5 days at room temperature, is also nice chilled, stored up to 5 days.
