Make the Churro Dough: In a saucepan, heat water, butter, sugar, salt, and sea salt over medium heat. Bring to a boil, stirring constantly. Reduce heat to low. Immediately add flour and stir vigorously until the dough forms a ball. Remove from heat, transfer to a bowl, cool for 5 minutes, then beat in the egg.
Make the Salted Caramel Sauce: In a small saucepan over medium heat, combine sugar and water. Cook without stirring until it turns a deep amber color. Remove from heat, carefully whisk in butter (mixture will bubble), then slowly whisk in heavy cream. Stir in vanilla and salt. Keep warm.
Heat Oil: Fill a deep fryer or pot with 2 inches of oil. Heat to 360°F (180°C).
Pipe and Fry Churros: Transfer dough to the pastry bag. Pipe 4-5 inch ropes of dough directly into the hot oil. Use a knife or scissors to cut the dough. Fry in batches until golden brown, flipping once.
Drain and Coat: Remove churros with tongs or a slotted spoon and drain on paper towels. Immediately roll in the cinnamon-sugar mixture.