Preheat the oven to 425°F.
Preheat the 10-inch cast-iron skillet on the stovetop on medium heat or in the oven as it preheats. I preheat mine in the oven since it's on and preheating anyway.
Combine eggs, milk, flour, sugar, and vanilla extract in a large bowl or a food processor and whisk/process for 1-2 minutes or until the batter is smooth.
Melt butter in the piping hot skillet and place it in the oven for 1-2 minutes, just long enough for the butter to turn golden brown.
Swirl the browned butter around the hot pan so the bottom is covered. Then pour batter directly into the hot, brown butter-coated skillet and return to the oven.
Bake in the preheated oven at 425°F for 18 minutes or until the top of the Dutch pancake is golden brown and it has puffed up quite a bit.
Slice and serve hot with melted butter, a drizzle of pure maple syrup, and powdered sugar, or any number of sweet toppings like fresh berries, lemon curd, or ice cream!