In a bowl, whisk the eggs just until barely combined. Whisk in the milk, buttermilk, superfine sugar, and vanilla, mixing only until barely blended.
Dump the pancake mix in a large bowl, make a well in the center, and pour in the egg mixture. Stir with a whisk until the batter is almost smooth. A few lumps are okay. Don't overmix the batter or the pancakes will be tough and rubbery. Let the batter rest a few minutes on the counter while you heat the skillet.
Heat a 9-inch crêpe pan or a nonstick skillet or a cast-iron skillet over medium heat. The skillet is ready when you dip a finger in cold water, flick the droplets onto the skillet, and the water immediately skitters and dances briefly before evaporating.
Brush the skillet with clarified butter to lightly coat, then wipe off any excess with a clean kitchen or paper towel. Using a 2-inch diameter ice-cream scoop or 1/3-cup measuring cup, pour a generous 1/3 cup batter into the skillet, tilting it from side to side and in a circular motion until the batter coats the bottom of the skillet. Cook only until the underside of the pancake begins to brown lightly, 1 to 2 minutes. Using a spatula, carefully lift the pancake and quickly flip it over. Cook until the underside is lightly brown, 1 to 2 minutes.
Serve immediately. (If you must, you can keep the pancakes warm on a baking sheet lined with parchment paper in a preheated 200°F (90°C) oven. You should be able to fit 2 in a single layer on the sheet. Place a sheet of parchment paper between each layer.) Slather the pancakes with softened butter and sprinkled with brown sugar, if desired. Repeat with the remaining batter.