Whisk sourdough starter, eggs, melted butter, and salt together in a bowl. Pour in milk and stir until batter reaches a thin, smooth consistency.
Heat a nonstick pan over medium-high heat. Add about 1 teaspoon cold butter and let melt. Pour in 1/4 cup batter and tilt pan in a circular motion to spread batter over the bottom. Cook until bottom is light brown, about 2 minutes. Carefully loosen and flip; cook for about 30 seconds more. Transfer to a plate. Repeat with remaining butter and batter.