Start by making the churro dough. In a medium saucepan, combine the water, butter, and salt. Heat over medium, stirring occasionally until the butter completely melts and the mixture comes to a rolling boil.
Once boiling, reduce heat to low and dump in all the flour at once. Stir vigorously with a wooden spoon until the mixture forms a smooth ball and pulls away cleanly from the sides of the pan. You’ll notice a thin film forming on the bottom of the pan.
Transfer the dough to a mixing bowl and let it cool for about 5 minutes. It should be warm but not hot – if it’s too hot, it’ll cook the eggs when we add them.
Add the eggs one at a time, beating well after each addition. The dough will look separated and glossy at first – keep mixing until it comes back together into a smooth, thick paste before adding the next egg. Mix in the vanilla extract last. The final dough should be thick and glossy, dropping slowly from the spoon in a V-shape.
Heat about 2 inches of oil in a heavy-bottomed pot to 375°F. While the oil heats, mix the sugar and cinnamon in a shallow dish. Also, line a plate with paper towels for draining.
Transfer the dough to a piping bag fitted with a large star tip. Pipe 3-inch straight lines of dough directly into the hot oil, using scissors to cut the dough. Don’t overcrowd – I usually fry 4-5 at a time. The churros will sink then float to the surface.
Fry for about 2-3 minutes per side, or until they’re a deep golden brown. They should look crispy and ridged on the outside. If they’re browning too quickly, your oil is too hot.
Remove with a slotted spoon and drain briefly on paper towels. While still hot, roll them in the cinnamon-sugar mixture. The coating sticks better when they’re warm.
Let the churros cool completely before assembling – about 30 minutes. Meanwhile, take your ice cream out of the freezer to soften slightly.
To assemble, slice each churro lengthwise, but don’t cut all the way through – leave one side connected like a book. Scoop about 1/4 cup softened ice cream onto one side and press gently to close. The ice cream should squish slightly out the ridges.
Place the assembled sandwiches on a baking sheet and freeze immediately for at least 1 hour until firm.