Preheat oven to 350°F (175°C). Position a rack in the center of the oven.
In a large bowl, whisk together 3 cups all-purpose flour, 1 1/2 cups granulated sugar, 1 teaspoon baking powder, and 1/2 teaspoon baking soda until evenly combined.
Cut in 1 cup (1 stick) butter until the mixture resembles small pebbles or coarse crumbs (use a pastry cutter, two forks, or your fingertips).
Make a well in the center of the dry mixture and set the bowl aside.
In a small bowl, whisk together 2 slightly beaten eggs, 1 cup buttermilk, and 1 teaspoon vanilla extract.
Pour the liquid mixture into the well of the dry mixture and stir just until moistened. The batter will be lumpy—do not overmix.
Spread half of the batter evenly into an ungreased 9 x 11-inch baking pan. Spoon the remaining batter into mounds over the first layer (do not smooth the top).
Bake for 30–40 minutes, or until the top is lightly golden and a toothpick inserted into the center comes out clean or with a few moist crumbs.
Remove from the oven and let the cake cool in the pan on a wire rack for at least 10–15 minutes before cutting and serving.