Steps

1. Combine the butter, ½ cup of the milk, and ½ cup water in a medium saucepan over high heat. Season generously with salt and pepper. Bring to a boil. When the butter has melted, remove the pan from the heat.
2. Add the flour all at once and stir vigorously with a wooden spoon until the mixture forms a thick dough and pulls away from the sides of the pan, 1 to 2 minutes. Return the pan to the heat for 1 minute, stirring constantly. Remove from the heat and let the dough cool to room temperature.
3. Beat in the eggs, one at a time, making sure each egg is completely incorporated into the mixture and the dough is smooth after each addition. The dough should be thick, shiny, and smooth. Add 1 cup of the cheese and beat in until well combined.
4. Preheat the oven to 400 degrees.
5. Spoon tablespoon-size mounds of the dough onto nonstick baking sheets, leaving about 1 inch between each. Brush the tops with the remaining ¼ cup milk, then sprinkle with the remaining ½ cup cheese.
6. Bake one tray at a time in the lower third of the oven until the gougères have doubled in size and are golden, 20 to 25 minutes. Serve warm or at room temperature.