Preheat oven to 375°F.
Bring water to a boil with the butter and salt in a 3-quart heavy saucepan over high heat, and then reduce to medium. Add flour all at once and cook, stirring vigorously with a wooden spoon until mixture pulls away from the side of the pan, about 2 minutes. Remove from heat and cool slightly, about 3 minutes, then add 4 eggs, 1 at a time, beating well after each addition. Batter will appear to separate, then become smooth. Mixture should be glossy and just stiff enough to hold soft peaks and fall softly from a spoon. If batter is too stiff, beat remaining egg in a small bowl and add to batter one teaspoon at a time, beating and then testing batter until it reaches proper consistency. Stir in cheese and dill.
Line 2 large baking sheets with parchment or lightly butter sheets. Fill pastry bag with batter and pipe 15 1-inch-diameter mounds (or spoon level tablespoons) 1 inch apart on sheets. Bake in upper and lower thirds of oven, switching position of sheets halfway through baking, until puffed, golden and crisp, about 30 minutes total. Make more puffs in same manner.
Makes about 60 puffs.