Make the churros: Line a baking sheet with parchment paper and fit a pastry bag with a 1/2-inch star tip; set aside. Combine 1 cup water, the butter, shortening, 1 tablespoon granulated sugar and the salt in a medium saucepan and bring to a boil. Stir in the flour with a wooden spoon, then reduce the heat to low and stir vigorously until the mixture forms a ball, about 1 minute. Let cool slightly, then transfer the dough to the pastry bag. Pipe 1-inch logs onto the prepared baking sheet, using kitchen shears to snip off each piece (you should have 26 to 28 churros). Let rest at room temperature, about 20 minutes.
Meanwhile, make the hot chocolate: Bring the milk to a simmer in a saucepan over medium heat. Add the chocolate, dulce de leche and cinnamon stick. Cook, stirring, until the chocolate melts and the mixture is combined, about 2 minutes. Cover and keep warm over low heat.
Fry the churros: Heat about 2 inches of vegetable oil in a Dutch oven or large pot until a deep-fry thermometer registers 360 degrees F. Combine the remaining 1/4 cup granulated sugar, the brown sugar and ground cinnamon in a large bowl. Working in three batches, fry the churros until golden and crisp, 4 to 5 minutes. Remove with a slotted spoon and transfer to a rack set over a baking sheet. Let cool about 5 minutes, then add to the cinnamon sugar and toss to coat.
Remove the cinnamon stick from the hot chocolate and divide among mugs. Top with whipped cream and drizzle with dulce de leche. Serve with the churros.