16 tablespoons (226 grams; 2 sticks) unsalted butter, at room temperature, plus more for greasing the pan
3 cups (360 grams) flour
1/2 teaspoon (3 grams) baking soda
1/2 teaspoon (3 grams) kosher salt or fine sea salt
2 cups (400 grams) sugar
3 large eggs, at room temperature
1/2 teaspoon vanilla extract
1 cup well shaken buttermilk, preferably full-fat
3 peeled, pitted peaches, sliced 1/2-inch thick (about 2 cups; see NOTE; can use frozen)
1 cup (150 grams) fresh or frozen blueberries
Preheat the oven to 350 degrees. Line a 9-by-13-inch baking pan with parchment paper so the 2 shorter sides overhang a bit (for lifting the cake out of the pan.) Grease the paper with a little butter.
Whisk together the flour, baking soda and salt in a medium mixing bowl.
Combine the 16 tablespoons of butter and the sugar in the bowl of a stand mixer; beat on medium speed, 3 to 4 minutes, until light and fluffy. Add the eggs one at a time, mixing well after each addition. Scrape down the bowl from time to time.
Add the vanilla extract to the buttermilk and stir to combine. On low speed, alternately add the flour mixture and the buttermilk mixture in two or three additions, ending with the flour, mixing until just barely incorporated. Use a flexible spatula to gently fold the batter a bit more by hand, making sure to scrape the bottom of the bowl to work in any residual dry ingredients. Once the batter looks combined with no white streaks, scrape it into the prepared pan. Smooth the top with an offset spatula.
Arrange peach slices on the top and scatter the blueberries evenly over the peaches. Bake (middle rack) for about 1 hour (start checking after 50 to 55 minutes), until the cake is golden brown on the edges and begins to pull away from the sides. During the baking, the batter will puff up over the fruit; once the cake cools, it will deflate a bit.
Transfer to a wire rack to cool completely (in the pan). Lift the cake using the parchment paper ends, then discard the paper and cut into 15 to 20 squares.
NOTE: To peel peaches, plunge them it into boiling water for 30 to 45 seconds, then use a slotted spoon to transfer them to a bowl of ice water. As soon as they are cool enough to handle, slip off the skins. The riper the peach, the less time it needs in the boiling water.