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In a large bowl, whisk together the flour, sugar, baking powder, baking soda, and salt.
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In a separate bowl or large measuring cup, whisk together the egg, buttermilk, butter or oil, and vanilla.
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Pour the wet ingredients into the dry ingredients, stirring to combine. Stir until the mixture is fairly smooth; some small lumps are OK.
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Allow the batter to rest, uncovered, for 15 minutes.
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While the batter is resting, heat a large skillet over medium heat or preheat a griddle to 350°F, until the surface is hot enough for a droplet of water to skitter across it. Lightly grease the pan with butter or vegetable oil.
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Spoon the batter, 1/4 cup at a time, onto the hot surface; a scone and muffin scoop works well here.
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Cook pancakes on the first side until bubbles form on the tops and the bottoms are brown, about 1 to 2 minutes. Flip and cook until the bottoms are brown, 1 to 2 minutes longer.
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Serve the buttermilk pancakes immediately, or hold briefly in a warm oven.
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Leftover buttermilk pancakes can be frozen the same day they're made and reheated in a 250°F oven.