Brown the butter in a small saucepan over medium heat, stir constantly until it turns golden and smells nutty, about 90 seconds past fully melted. Pour into a bowl immediately to stop cooking.
Whisk flour, baking powder, baking soda, sugar, and salt in a large bowl. Make a well in the center.
In a separate bowl, whisk buttermilk, egg yolks, browned butter, sour cream, and vanilla until combined.
In a third bowl, whip egg whites with a hand mixer or whisk until soft peaks form, they should hold a gentle curve but not be stiff.
Pour wet ingredients into the flour well. Fold 8–10 times only, batter should still be lumpy. Stop before you think you should.
Gently fold in whipped egg whites in two additions, use a light hand, you're preserving air bubbles not mixing them out.
Rest the batter for 5 minutes, this lets the baking powder activate and the gluten relax. Don't skip this.
Heat a cast iron skillet or non-stick pan over medium heat. Brush lightly with butter.
Pour ¼ cup batter per pancake. Don't spread it, let it settle naturally.
Cook until bubbles form across the surface and edges look set, about 2–3 minutes. Flip once only.
Cook second side 1–2 minutes until golden. Never press down with the spatula, you're deflating every air bubble you just worked to create.
Serve immediately or keep warm in a 200°F oven on a wire rack, never stack on a plate or the steam makes them soggy.