Prepare your ice cream maker—some manufacturers require you to freeze your insert for at least 24 hours.
In a large saucepan, mix together the evaporated milk, heavy cream, ¾ cup of sugar, salt, and corn syrup. Stir well and bring to a boil while stirring occasionally.
In a bowl, mix together the remaining ¼ cup of sugar, yolks, and cornstarch.
Once the milk mixture reaches a boil, turn the heat down to low.
Slowly drizzle one cup of the hot milk into the egg yolks as you stir the yolk. Drizzle in a second cup of hot milk.
Transfer the yolk mixture to the saucepan and cook until the mixture is thick enough to coat the back of a wooden spoon,
Strain through a fine mesh strainer into a gallon freezer bag. Seal and chill completely—at least a few hours.
While the mixture is cooling, start on the swirl. In a saucepan, add all the ingredients together and heat on medium. Stir constantly, while scraping the bottom and sides of the pan to prevent burning. Keep cooking until the mixture is about halved in volume and is a nice caramel color. Transfer to a bowl and chill.
Once the ice cream base is thoroughly chilled, begin churning according to manufacturer instructions (you may have to churn in two batches). Once the ice cream is almost finished, add in the vanilla, chop the churros, and add them in (if you are doing it in batches, divide the vanilla and chopped churros accordingly).
Begin adding the ice cream to your container, swirling in some of the cinnamon dulce de leche as you go. Freeze until firm.