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1
Heat oil in Dutch oven or large pot over medium-high heat until temperature registers 350u00b0F on deep-fat thermometer. (Oil should be 1- to 1 1/2 inches deep).
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2
Fit pastry bag or sturdy gallon-sized zipper-lock bag with 1/2-inch wide star tip (if using plastic bag, snip off one corner of bag to fit the tip); set aside.
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3
Line a large baking sheet with two layers of paper towels. Combine all but 1 tablespoon sugar and cinnamon in large plate; set aside.
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4
Bring milk, butter, salt, remaining 1 tablespoon sugar, cinnamon stick, vanilla, and nutmeg to boil over medium-high heat in large heavy-bottomed saucepan. Turn off heat and let stand 10 minutes to allow flavors to infuse milk.
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5
Discard cinnamon stick, bring milk mixture back up to boil over medium-high heat, then add flour and beat vigorously with wooden spoon until a smooth dough forms.
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6
Transfer dough to large bowl and let cool 5 minutes. Add eggs, one at a time, beating each until fully incorporated before adding the next (eggs may be beaten into dough with electric mixer or by hand with wooden spoon).
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7
Scrape dough into prepared pastry bag. Carefully pipe dough out onto hot oil into 6-inch long churros, pinching the dough off the star tip with thumb and index finger. Fry batches of 4 to 5 churros for 3 to 4 minutes until deep golden brown.
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8
Transfer churros with tongs to prepared baking sheet just to blot off excess oil, then transfer to cinnamon-sugar plate and roll to coat.
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9
Place dulce de leche, chocolate chips, salt, and butter in medium bowl and microwave 1 to 2 minutes, until chips are melted, pausing microwave to stir mixture every 20 seconds. Serve with warm churros.
