In a small bowl, stir together ¼ cup warm water and yeast; let stand until mixture is foamy, about 5 minutes.

In the bowl of a stand mixer fitted with the paddle attachment, combine flour, granulated sugar, and salt. With mixer on low speed, add yeast mixture, milk, sour cream, egg, butter, and vanilla, stirring just until combined. Switch to the dough hook attachment; beat at medium speed until smooth and elastic, about 5 minutes.

Spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. Cover and refrigerate for at least 6 hours or up to 24 hours.

Fill a medium Dutch oven with canola oil to halfway full, and heat oil over medium heat until a deep-fry thermometer registers 350°. On a lightly floured surface, roll dough ¼ inch thick. Using a sharp knife, cut dough into 2½ inch squares. Working in batches, fry dough, turning frequently, until golden brown, 2 to 3 minutes. Remove from oil, and let drain.