Cut twelve 4-inch squares of parchment paper and set aside. These will help transfer crullers to hot oil.

In a medium saucepan, combine water, butter, sugar, and salt. Bring to a rolling boil over medium-high heat.

Remove from heat and add flour all at once, stirring vigorously with a wooden spoon until mixture forms a smooth ball that pulls away from sides of pan.

Transfer dough to a mixing bowl and let cool for 5 minutes. Add eggs one at a time, beating well after each addition until dough is smooth and glossy.

Stir in vanilla extract. Transfer dough to a piping bag fitted with a large star tip.

Pipe dough into 3-inch rings on parchment squares, overlapping ends slightly. Refrigerate piped crullers for 15 minutes.

Heat vegetable oil in a large pot to 375°F. Carefully place crullers in oil parchment-side up, then remove parchment with tongs.

Fry crullers for 2-3 minutes per side until deep golden brown. Remove to wire rack and cool completely.

For glaze, melt butter in saucepan over medium heat. Whisk in powdered sugar, rum, milk, cinnamon, nutmeg, and vanilla until smooth.

Dip cooled crullers in warm glaze, allowing excess to drip off. Place on wire rack set over baking sheet to catch drips.

Let glaze set for 10-15 minutes before serving. Serve immediately for best texture and flavor.