Whisk the gluten free flour, tapioca starch, baking powder, sugar, salt and cinnamon together in a bowl to combine. Heat the milk with the butter in a saucepan. Bring to a boil and remove from the heat.
Pour into the flour mixture and stir until well combined, or use a mixer to combine completely. Allow to cool for five minutes.
Add the eggs, one a time, and vanilla and hand stir vigorously or mix with a mixer on medium until dough is smooth, fairly thick and sticky. Add additional milk to thin if needed (1 tablespoon) until dough is a piping consistency. Cool completely or refrigerate for 1 hour.
At this point, you can proceed with cooking the churros or refrigerate the dough for up to 24 hours. If you refrigerate the dough for more than a few hours, before frying, bring dough to room temperature for 30-45 minutes before proceeding to fry.
When ready to fry your churros, over medium high or high heat, in a large saucepan or Dutch oven, approximately two quarts of oil to 350 degrees.
Fit a pastry bag with a 1M star tip (or what you have that is larger than 1/4″ inch wide) and fill the bag with the churro dough. Pipe dough directly into the hot oil, approximately 6-7" long, carefully, and two or three max at a time.
Fry for 1-2 minutes per side, or until golden brown, before removing with tongs to a wire rack seated over a plate covered with several layers of paper towels.
Let cool until you can handle. Then dip in granulated sugar or cinnamon sugar, until well coated. Or serve with thick dipping chocolate, recipe below, if desired, or use a brand of a packaged type of thick chocolate from Spain (which is gluten free).
After churros are cooked and very slightly cooled to handle easily, dip each one in the cinnamon sugar mixture.