In a medium saucepan, bring water, butter, sugar, and salt to a boil over medium heat.
Once boiling, remove from heat and add the flour all at once, stirring vigorously until the mixture forms a smooth dough.
Allow the dough to cool slightly for about 5 minutes, then mix in the eggs, one at a time, ensuring each is fully incorporated before adding the next.
Stir in vanilla extract, nutmeg, and half of the cinnamon, blending well until smooth and glossy.
Heat oil in a deep frying pan to 375°F (190°C).
Transfer the churro dough to a piping bag fitted with a star tip.
Once the oil is hot, carefully pipe 4 to 6-inch long strips of dough into the oil, cutting with scissors to release the dough into the oil. Fry until golden brown, about 2-3 minutes per side.
Remove churros and drain on paper towels. While still warm, toss in a mixture of granulated sugar and the remaining cinnamon (and a pinch of nutmeg if desired).
Optionally, dust with powdered sugar before serving for an extra festive touch.
Notes
Arrange the churros on a serving plate, and serve with a small bowl of hot chocolate or eggnog for dipping. Garnish with a sprinkle of nutmeg on top.