In a medium saucepan, stir 1-cup water, sugar and salt together. Bring to a boil over medium heat. Remove pan from heat and add flour to the mixture all at once. Quickly beat ingredients together with a wooden spoon to blend completely. Return pan to medium-low heat and continue beating until mixture clings together and a film forms on bottom of pan. Remove from the heat. Break egg into dough and quickly beat until egg is absorbed and mixture is smooth.
In a medium, heavy-bottomed saucepan, pour oil to a depth of about 2 inches and heat over medium-high heat to 350° to 400°F. Test oil by adding a small piece of dough. It should sizzle immediately when placed in oil.
In a shallow bowl, combine sugar and cinnamon and set aside.
Place dough in a pastry bag fitted with #6 star tip. Force a 4-inch length of dough through the tip directly into the hot oil. Fry until golden, drain on paper towels. Roll churro in cinnamon sugar. Repeat until dough is used up (you should have 12 churros).
Top each serving (about 3 churros) with cajeta and a small scoop of ice cream.