Preheat oven to 200°F (93°C). Melt 6 tablespoons unsalted butter and set aside to cool slightly.
In a large bowl, whisk together 2 1/4 cups all-purpose flour, 1/3 cup sugar, 1 1/4 teaspoons baking powder, 1/2 teaspoon baking soda, and 1/4 teaspoon salt until evenly combined.
In another large bowl, whisk together 2 large eggs, 2 cups buttermilk, and 1 teaspoon vanilla extract until smooth. Pour in the slightly cooled melted butter and whisk until fully incorporated.
Pour the wet mixture into the dry mixture. Gently whisk until just combined. Stop as soon as no large patches of dry flour remain. The batter should be thick with a few small lumps. Do not over-mix.
Place a griddle or large skillet over medium heat and let it heat for about 2 minutes. Test with a drop of water; if it sizzles and dances, it's ready. Generously coat with butter or nonstick spray.
Drop or pour a heaping 1/4 cup of batter onto the hot griddle for each pancake. Cook undisturbed until edges are set and bubbles appear on surface, about 2 minutes. Flip carefully and cook until golden brown and cooked through, about 1 to 2 more minutes.
Transfer finished pancakes to a baking sheet in preheated 200°F oven to keep warm while cooking remaining batter. Add more butter to griddle between batches as needed.
Serve immediately with favorite toppings. Cover and store leftovers in refrigerator for up to 5 days.