Set a wire rack inside a baking sheet; set aside.
In a medium bowl, whisk together buckwheat flour, all-purpose flour, sugar, salt, baking powder, and baking soda; set aside. In a separate bowl, whisk together buttermilk, milk, egg, and melted butter. Whisk buttermilk mixture into four mixture just until combined and small pockets of dry flour remain.
Heat a large nonstick skillet over medium heat for 3 minutes. Add a small pat of butter (about 1 teaspoon) to skillet and swirl to coat.
Spoon dollops of batter, scant 1 tablespoon each, evenly around skillet (about 6–8 pancakes at a time).
Cook blini until large bubbles form on surface and bottoms are golden brown, 1½–2 minutes. Flip blini and cook until golden brown on second side, about 1½ minutes more.
Transfer blini to prepared sheet. Wipe skillet clean with paper towels. Continue to repeat the cooking process with additional butter and remaining batter, wiping skillet clean after each batch.
Serve as desired (see instructions below for how to keep blini warm while cooking all of the blini—should you choose to serve them warm).