Stir the warm whole milk, yeast, and 1 tablespoon of the granulated sugar together in a large bowl. Let the mixture stand for 5 to 10 minutes, until visibly foamy.
Whisk in the remaining granulated sugar, Greek yogurt, melted butter, vanilla extract, salt, and nutmeg until smooth.
Add the flour gradually and mix until a soft, slightly tacky dough forms. Knead for 6 to 8 minutes, until the dough becomes smooth and elastic.
Place the dough in a lightly greased bowl, cover, and let it rise in a warm place for 60 to 90 minutes, until approximately doubled in size.
Roll the dough to about 1/4-inch thickness and cut it into 2 1/2-inch squares. Cover the pieces loosely and let them rest for 15 minutes while the oil heats to 350°F.
Fry the beignets in small batches for about 1 to 2 minutes per side, until puffed and deep golden. Transfer them to a paper towel-lined tray and allow excess oil to drain briefly.
Dust the warm beignets very generously with powdered sugar and serve immediately while the centers are soft and fluffy.
