Prepare the ice cream: Stir together cream and milk in a saucepan. Cook over medium, stirring occasionally, until just steaming, 5 to 6 minutes. Remove from heat.
Pulse chocolate in a food processor until finely chopped, 4 to 6 times. Add corn syrup, salt, half of milk mixture, and, if desired, mezcal to chocolate; process until smooth, about 1 minute.
Place remaining milk mixture in a large bowl; add processed chocolate mixture. Whisk until smooth. Chill over an ice bath or in refrigerator until cold, about 1 hour. Pour mixture into freezer container of a 1 1/2-quart ice-cream freezer; proceed according to manufacturer's instructions. (Instructions and times will vary.) Transfer to an airtight, freezer-safe container; freeze until firm, about 2 hours.
Prepare the churros: Heat oil in a large, wide Dutch oven or cast-iron skillet over medium-high to 360°F.
Stir together cinnamon and 1/4 cup of the sugar in a shallow dish, and set aside. Combine water, butter, salt, and remaining 1/2 cup sugar in a large saucepan. Cook over medium-high until butter is melted, 3 to 4 minutes. Add flour and stir continuously with a wooden spoon or stiff rubber spatula until very smooth and a ball forms, about 1 minute. Remove from heat.
Add eggs, 1 at a time, stirring until completely blended after each addition. Transfer mixture to a pastry bag fitted with a large star tip. Pipe 3- to 4-inch churros into hot oil, using a paring knife to cut to desired length. Fry until golden brown and crispy, 5 to 6 minutes, turning occasionally. Drain well on paper towels. Dredge churros in cinnamon mixture while still warm. Serve with ice cream.
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