In a large skillet, heat 1 & 1/2 inches of vegetable oil over medium/medium-high heat and stick in a candy or deep fat thermometer, heating until it reaches 360 degrees F.
In a loaf pan or small bowl, whisk together the cinnamon sugar coating until combined; set aside.
In a medium skillet, combine the cubed butter, water, granulated sugar and salt. Bring to a boil over medium-high heat, then add in the flour and reduce heat to medium-low, cooking and stirring with a rubber spatula until combined and a soft dough forms. It's okay if the dough has some lumps in it. Allow the dough to cool off the heat for 5 minutes.
Place dough in the bowl of a stand mixer. Add in the egg and vanilla and beat on medium-high speed until combined and cohesive. Be careful with this step as the hot dough plus a cold egg could easily scramble your egg, so make sure you work quickly! Place the dough into a disposable piping bag attached with a star or French star tip.
Pipe 4 to 5-inch long strips of churro dough into the hot oil, using scissors or a knife to cut off each piece. Fry 4 to 5 at a time, gently separating them with tongs if they attach to one another. Fry for about 2-3 minutes per side or until golden brown and crisp. Drain on a paper towel-lined plate for 15-20 seconds, then dredge in the cinnamon sugar mixture.
Serve warm with dulce de leche dipping sauce!