Set the oven to 350°F (175°C). Prepare a 9×13-inch baking pan by greasing it lightly or lining it with parchment paper for easier removal.
In a large bowl, sift or whisk together the flour, granulated sugar, baking powder, baking soda, and salt to combine thoroughly.
In a separate mixing bowl, beat the eggs briefly. Incorporate the buttermilk, cooled melted butter, and vanilla extract, stirring until the mixture is well blended.
Pour the wet ingredients into the dry mixture and stir gently until just combined. Do not overmix to preserve the cake's soft texture.
Carefully fold 1¼ cups of the blueberries into the batter, reserving the remainder for the topping.
Transfer the batter to the prepared pan, smoothing it out into an even layer. Sprinkle the reserved blueberries across the top and, if using, scatter the coarse sugar over the surface.
Place in the oven and bake for 35 to 40 minutes, or until a toothpick inserted into the center emerges clean.
Let the cake cool in the pan for at least 10 minutes before cutting into portions. Serve either warm or at room temperature.