Heat the oven to 400°F (205°C). Line a large baking sheet with parchment paper.
Whisk the flour, cornmeal, sugar, baking powder, salt, and nutmeg together in a large bowl until evenly combined.
Rub the softened butter into the dry ingredients with clean fingertips until the mixture resembles coarse crumbs.
Stir in the milk and vanilla just until a soft dough forms. If dry flour remains, add the extra tablespoon of milk. Do not overmix.
Turn the dough onto a lightly floured surface and gently knead it 3 or 4 times, only until smooth. Divide it into 10 equal pieces.
Roll each piece into a short log about 4 inches (10 cm) long, with slightly tapered ends. Arrange the pieces on the prepared baking sheet, leaving space between them.
Bake until puffed, golden brown, and firm to the touch, 16 to 18 minutes. Cool on the baking sheet for 5 minutes before serving warm.