The batter can also be made in a blender: Mix the eggs until pale and frothy, then add the remaining ingredients and puree until smooth.
Preheat oven to 450 degrees with a 10-inch cast-iron skillet on center rack. Melt butter in a medium saucepan over medium-high. When it boils, reduce heat to medium; simmer until foamy. Continue cooking, stirring occasionally and scraping bottom of pan, until foam subsides, butter turns golden brown with a nutty aroma, and milk solids separate into brown specks that sink to bottom, 2 to 7 minutes. Remove from heat; immediately transfer to a heatproof bowl.
Stir together sugar and cardamom. In a large bowl, beat eggs on medium-high until pale and frothy, about 2 minutes. Add flour, salt, milk, vanilla, and 1 tablespoon cardamom sugar; beat until smooth, about 1 minute (batter will be thin).
Carefully remove skillet from oven. Add brown butter and apple, stirring a few times to evenly coat. Spread apple into a single layer; pour batter into skillet. Bake until puffed, crisp on top, and golden brown in places, 20 to 25 minutes. Immediately sprinkle with 2 tablespoons cardamom sugar. Cut into wedges and serve, with sour cream and remaining cardamom sugar.