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Heat oil in a pan to 350 F.
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In a medium saucepan, combine milk, water, butter, sugar, and salt. Bring to a boil, and immediately remove from heat, stirring in vanilla, eggs, and flour until dough forms. Transfer to a bowl to cool.
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Combine sugar and cinnamon in a shallow bowl.
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Once cooled, transfer the dough into a pastry bag, and pipe one-inch pieces of dough, cutting with scissors, into the hot oil. Cook until golden brown, about 3 minutes.
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Toss the warm churro bites in the cinnamon-sugar mixture.