Combine cinnamon and sugar in a wide shallow bowl and stir until well mixed. Set aside.
Fill a large, heavy-bottomed pot 2 inches deep with oil. Attach a thermometer to the pot and heat oil over medium-high heat to 360°F.
While oil heats, make churro dough. Melt butter, 1 cup water, sugar and salt in a medium saucepan over medium heat. Bring to a boil then reduce heat to low. Add flour all at once and stir until completely mixed. Cook, stirring constantly, until a smooth dough forms and pulls away from the sides of the pot, about a minute. Transfer to a mixing bowl and let cool 5 minutes.
Use a handheld mixer to beat in the eggs, one at a time, until the dough is smooth and shiny. Transfer dough to a piping bag fitted with a ¼” star tip and set aside.
When oil is hot, carefully pipe a 6-inch line of batter into the oil and snip the end with kitchen shears. Fry 2-3 at a time for about 3 minutes per side, or until golden brown and cooked through. Drain on a paper towel-lined plate then roll in cinnamon sugar mixture while hot. Repeat frying, draining and coating the remaining churro batter. Serve warm with salted caramel sauce.
Heat sugar in a medium saucepan over medium heat. Cook, stirring with a rubber spatula, until melted and dark amber in color. Sugar will look crumbly at first; keep stirring. Reduce heat if needed, so all of the sugar clumps can melt without syrup getting too dark.
Whisk in butter until completely melted. Slowly pour in heavy cream (it will bubble up rapidly) and whisk until smooth.
Gently simmer for about 1 minute, whisking to melt all of the caramel and produce a smooth sauce. Stir in vanilla extract and flaky salt. Caramel will thicken as it cools.
Transfer to a bowl and serve warm, with churros.
Store salted caramel sauce, covered and refrigerated, for up to 1 month. Gently reheat in the microwave or stove.