Mix together 3 cups of the flour and the remaining ingredients until a shaggy dough forms. Dust a countertop with flour and knead the dough, working in the remaining 1/2 cup of flour as needed, until you get a smooth and supple ball of dough.
Place dough in a greased bowl, cover, and let rise for about 2 hours or until doubled.
Punch down dough and roll it out on a floured countertop to about 1/4 inch. Cut out doughnuts, and place onto a parchment-lined baking sheet. Gather scraps, reroll, and cut until finished.
Cover doughnuts and let rise 45 minutes to an hour.
Bake at 350°F for 15-18 minutes, until golden.
Notes
You might need more or less flour depending on your climate and the humidity level; start off with a lower amount if you aren't sure.