In a large bowl, whisk together the flour, cornmeal, sugar, baking powder, salt, and nutmeg until evenly combined.
Stir the milk and vanilla together. Pour the liquid into the dry ingredients and mix with a spoon or clean hands until a soft, slightly tacky dough forms. Add milk 1 tablespoon (15 ml) at a time only if there are dry, crumbly areas. Do not overwork the dough.
Let the dough rest for 5 minutes. Meanwhile, add about 2 inches (5 cm) of oil to a deep, heavy skillet or Dutch oven and heat it to 350°F (175°C). Line a plate or baking sheet with paper towels or a wire rack.
Divide the dough into 10 equal portions. With lightly oiled hands, roll each portion into a smooth log about 3 inches (8 cm) long, tapering the ends slightly if desired.
Carefully lower a few dough pieces into the hot oil, leaving room between them. Fry in small batches, turning occasionally, until deep golden brown on all sides and cooked through, about 4 to 6 minutes per batch. Keep the oil near 350°F (175°C).
Transfer the fried festival dough to the prepared rack or paper towels. Let cool briefly before serving warm.