Dissolve yeast in warm water in a large bowl. Let stand until yeast softens and begins to form a creamy foam, about 5 minutes.
Add milk, sugar, egg, oil, vanilla extract, and salt to the yeast. Vigorously beat in flour. Mix until a sticky dough forms.
Divide dough into quarters. Place 1 portion on a heavily floured surface. Roll to about 1/8-inch thickness using a well floured rolling pin. Cut into 2-inch squares. Repeat with remaining 3 portions.
Heat oil in a deep-fryer or large saucepan to 370 degrees F (188 degrees C). Fry beignets, a few at a time, until they puff and rise to the top, 2 to 3 minutes. Turn and brown the other side, 1 to 2 minutes. Drain on paper towels.
