In a medium-sized saucepan, combine 1 cup of water, 1/4 cup of diced unsalted butter, 2 tablespoons of white granulated sugar, 1/4 teaspoon of cinnamon, 1/4 teaspoon of nutmeg, and 1/4 teaspoon of salt. Bring this mixture to a boil over medium heat, stirring until the butter has completely melted.
Once your mixture is boiling, remove it from the heat and quickly stir in 1 cup of all-purpose flour. Mix vigorously until the dough forms a ball and pulls away from the sides of the pan.
Let the mixture cool for a few minutes, then add 1 cold large egg and 1 teaspoon of vanilla extract. Beat the dough until it is smooth and well combined.
In a deep fryer or heavy-bottomed pot, pour in enough vegetable oil to submerge the churros (about 2-3 inches deep). Heat the oil over medium-high heat until it reaches 350°F (175°C).
While the oil is heating, transfer your churro dough into a piping bag fitted with a large star tip. Once the oil is hot, carefully pipe strips of dough directly into the oil, cutting them to your desired length with scissors.
Fry the churros for about 2-3 minutes per side or until they turn a beautiful golden brown. Use a slotted spoon to remove them from the oil and place them on a cooling rack.
In a shallow bowl, mix together 1/2 cup of white granulated sugar and 1 1/2 teaspoons of ground cinnamon. While the churros are still warm, roll them in this mixture until they are well coated.