In a large bowl, sift flour and set aside. Heat butter in the microwave until melted. Let sit for 4 minutes to slightly cool. Heat milk in the microwave for 1 minute until 110°F. Too hot or too cold milk can either kill or slow the yeast.
Use a stand mixer, and mix salt, sugar, yeast, vanilla, beaten egg, milk, and butter together. Switch to a dough hook and add in all of the flour.
Mix on low until a dough forms. Switch to medium and let it knead the dough for 5 minutes. Halfway through check to see how sticky the dough is. It should be tacky and not sticking to your fingers. Add ⅛ cup of flour at a time until it is no longer sticky.
Place dough ball into a large mixing bowl sprayed with nonstick spray. Cover with cling wrap and let it rise for 1 hour. Transfer to the fridge and let chill overnight.
Roll the dough out on a lightly floured surface until it is ⅛-1/4 inch thick. Use a doughnut cutter (about 3 ½ inch diameter biscuit cutter) and cut out doughnuts. You can re roll the dough once to get more doughnuts. Save the centers of the doughnuts for doughnut holes.
Place the doughnuts on a cutting board or a sprayed cookie sheet. Wrap in cling wrap. Let rise for 2 hours.
Heat oil in a deep pot over medium. Heat until oil is 350°F. Use a thermometer to watch the temperature the whole time. It should be right at 350°F.
Fry 2 doughnuts at a time. Cook them 1 minute per side. Heat until each side is lightly golden. I used a metal spatula to flip them. Once done. Try to drip off as much oil as you can. Place on a cooling rack lined with paper towel to soak up the oil. Let the oil get back to 350°F before frying the next batch.