Combine milk, yeast, sugar and vanilla in the bowl of your stand mixer, or in a large bowl, and let sit for 10 minutes. Mix in the eggs, 1/4 cup butter and salt.
Add the 3 1/2 cups flour to the yeast mixture and mix until a sticky dough is created. Grease the inside of a large bowl with the 1 Tbsp. butter. Lightly flour a work surface. Remove dough from the bowl and set on the work surface. Knead and shape dough into a ball. Put dough into the bowl and cover with plastic wrap. Let dough rise in a warm place until doubled in size, about 90 minutes. Preheat the oil in your deep fryer to 325 degrees F. Combine the icing sugar and spices on a wide plate. Transfer the dough back to the floured surface. Roll or press out into a large rectangle that’s about ¼” thick. Cut the dough into 2” squares. Deep-fry the squares, in batches, flipping a few times, until rich golden and cooked through, about 4 minutes. Drain the beignets well. Coat the beignets in the icing sugar mixture, and then set on a serving platter. Repeat with remaining squares of dough. Beignets are best enjoyed soon after cooking.