Preheat oven to 425 degrees F (220 degrees C). Line a baking sheet or baking stone with a silicone liner or parchment paper.
Place water, butter, sugar, and salt in a saucepan over medium-high heat. Cook, stirring occasionally, until mixture is melted and dissolved, about 5 minutes. Reduce heat to medium. Add flour all at once and cook, stirring vigorously, until a ball of dough forms and pulls away from the sides of the pan, 15 to 30 seconds.
Transfer dough to the bowl of a stand mixer fitted with a paddle attachment. Let cool, about 5 minutes. Add eggs one at a time, beating on medium-low speed after each addition, until batter is smooth and shiny, about 5 minutes.
Transfer batter to a pastry bag fitted with a large round tip. Pipe the batter into 2-inch mounds by squeezing close to the sheet and slowly pulling up.
Increase oven temperature to 450 degrees F (230 degrees C).
Bake in the preheated oven until puffy, about 10 minutes. Reduce oven temperature to 350 degrees F (175 degrees C) and continue baking until shells are crispy and set, 13 to 15 minutes more.
Place goat cheese, dill, and salt in a stand mixer fitted with a whisk attachment. Mix until filling is creamy and smooth, 2 to 3 minutes. Transfer to a pastry bag fitted with a small round tip. Pipe filling into the bottom of each shell until filled.