Beat eggs in a large bowl until smooth. Add milk, flour, sugar, vanilla extract, nutmeg, and salt; whisk until just combined. Stir in 1/2 cup melted butter until batter is very thin and slightly lumpy.
Heat a nonstick griddle over medium-high heat. Brush generously with 1 teaspoon butter. Pour batter onto the griddle to make 6-inch pancakes. Cook until bubbles form and the edges are dry, 30 seconds to 1 minute. Flip and cook until the other side is dry, 30 seconds to 1 minute. Repeat with remaining batter.