Make the dough. In a medium saucepan, combine water, butter, sugar, and salt over medium heat until butter melts and mixture simmers. Reduce heat to low, add flour all at once, and stir vigorously until dough forms a ball and pulls away from sides. Cook 1-2 minutes more to dry the dough.
Cool and add remaining ingredients. Remove from heat and cool 5 minutes. Stir in sourdough discard and vanilla until combined. Beat in eggs one at a time until dough is smooth and shiny. Cover and refrigerate overnight if desired for enhanced sourdough flavor.
Heat oil. Heat 2 inches of oil in deep skillet or Dutch oven to 350°F. Mix cinnamon and sugar in a small bowl. Line plate with paper towels.
Fry churros. Transfer dough to piping bag fitted with large star tip. Pipe 1 to 1.5 inch pieces into hot oil, cutting with butter knife. Fry in batches for 90 seconds, flip, and fry 30-60 seconds more until golden brown and puffed. Transfer to paper towels.
Coat and serve. While warm, toss churro bites in cinnamon sugar mixture. For chocolate sauce, microwave chocolate chips, cream, and cayenne for 30 seconds, then stir until smooth. Serve churros warm with sauce for dipping.