In the bowl of a stand mixer fitted with the paddle attachment, combine the flour, sugar, yeast and salt. In a pan, warm the milk until it registers 115° on an instant-read thermometer. Transfer to a bowl; whisk in the egg and vanilla. Add the milk mixture and 6 tbsp. butter to the flour mixture; mix on low until blended. Using the dough hook, knead on medium until shiny and tacky, 7 minutes.
Butter a large bowl. Add the dough; cover tightly with plastic. Let rise in a warm area until doubled in size, about 45 minutes. Line one baking sheet with parchment; fit another with a wire rack.
On a floured work surface, turn out the dough and sprinkle with flour. Pat into a 10-by-12-inch rectangle.
In a 6-qt. pot, heat the oil over medium-high until it registers 350° on a deep-fry thermometer. Using a 1 1/2-inch round cutter, cut out rounds; transfer to the parchment-lined sheet.
Using a slotted metal spoon, transfer five rounds to the oil. Fry, turning once, until puffed and golden, 2 to 3 minutes. Transfer the doughnuts to the rack; repeat with the remaining dough.