6cupsoil for fryingor enough to fill a large skillet with 1 1/2-2 inches of oil
To Make the Horchata Ice Cream:
Combine the rice and cinnamon stick in a blender. Blend on medium-high speed for 1-2 minutes, until the rice and cinnamon are finely ground. (It's okay if your blender produces a coarser mixture—you just want to break them down a little.) Add the cup of milk and blend the milk with the rice until you have a thick, slushy mixture. Pour it into a large container and add the cream, condensed milk, salt, and vanilla. Whisk well, then refrigerate it overnight, or at the minimum 3-4 hours.
Line a strainer with 4 layers of cheesecloth, and pour the ice cream mixture through the cheesecloth to strain out the rice and cinnamon stick. Churn it in an ice cream maker according to the manufacturer's directions. When it is finished it will probably have a soft-serve consistency, so place it in the freezer to firm up before scooping and assembling the sandwiches.
To Make the Churros:
Sift together the flour and 1 teaspoon cinnamon, and set aside for a moment.
Combine the water, sugar, vegetable oil, and salt in a medium saucepan over medium-high heat. Bring it to a boil, and once boiling, add the flour mixture all at once and stir it in quickly. It will be thick. Continue to stir until the streaks of flour are gone, and the whole thing comes together in a rough, shaggy ball in the middle.
Transfer the hot ball of dough to the large bowl of a stand mixer fitted with a paddle attachment. Begin to beat on medium speed, and add one of the eggs. At first it won’t come together, but continue to beat until it’s incorporated and the mixture smooths out. Add the other egg and beat until your dough is combined and smooth.
Fit a pastry bag with your star tip, and fill the bag with churro dough. Cover two baking sheets with parchment or waxed paper. Pipe the dough in a spiral shape into 3-inch circles, keeping the tip close to the baking sheet so the dough is in a very thin layer. You should get about 18-20 circles from one batch.
Transfer the sheets to the freezer for 10 minutes to chill the dough and make it easier to handle. While the churros are freezing, heat the oil for frying. Pour the oil into a large skillet, so that it’s 1 1/2 – 2 inches deep, and turn the heat to medium. Insert a candy/deep fry thermometer and let the oil heat to 350 F (175 C). Line a baking sheet with several layers of paper towel, and toss the granulated sugar and remaining tablespoon of cinnamon together in a shallow pie pan.
Once at 350 F, slide a spatula under the frozen churros to release them from the pan. Transfer 2-3 churros to the hot oil and fry them, turning them occasionally so that they brown on all sides.
When they're finished, they will be puffed and golden brown on all surfaces. Transfer them to the paper towel-lined baking sheet using a slotted frying spatula, then quickly move them to the cinnamon-sugar bowl and toss them with the sugar until they’re coated. Let the oil come back up to 350 F before frying the next batch. If it gets too hot during the cooking process, remove it from the heat until it cools down again. The goal is to always keep it around 350 F.
Continue to cook the churros in small batches until they're all fried and coated with cinnamon sugar. While the churros can be made a few hours in advance, for the best taste and texture, they should be enjoyed on the same day they’re made. To make the ice cream sandwiches, top one with a scoop of ice cream, place another churro on top, and serve immediately!
For the best flavor, I recommend letting the ice cream base sit overnight before churning it. If you don't have that time, it should sit for at least 3 hours to really let the flavors infuse. If you don't want to make ice cream at all, any store-bought variety can be substituted instead!