1. Grease a large bowl with cooking spray and set aside. In a small, microwave-safe bowl or glass measuring cup, add milk. Microwave until lukewarm, 40 seconds. Add a teaspoon of sugar and stir to dissolve, then sprinkle over yeast and let sit until frothy, about eight minutes.
2. In a medium bowl, whisk together flour and salt. In a large bowl, whisk together remaining sugar, butter, eggs, and vanilla with a wooden spoon. Pour in yeast mixture, mix to combine, then add in dry ingredients, stirring with a wooden spoon until a shaggy dough forms. Transfer to a lightly floured surface and knead until elastic and only slightly tacky, adding more flour a teaspoon at a time if needed, about five minutes.
3. Form into a tight ball, then place dough in oiled bowl and cover with a clean dish towel. Let dough rise in a warm spot in your kitchen until doubled in size, about one hour.
4. Line a large baking sheet with parchment. Punch down dough, then turn onto a lightly floured work surface and roll out into a 1/2-inch-thick rectangle. Using a doughnut cutter or a three inch biscuit cutter, punch out your doughnuts. Re-knead scraps together and punch out as well. Place doughnuts onto baking sheets, cover with dish towel, and let rise again, about 40 minutes.
5. Line a large baking sheet with paper towels. In a large Dutch oven over medium heat, heat two inches oil to 350 degrees Fahrenheit. Cook doughnuts, in batches, until deeply golden on both sides, about one minute per side.
6. Transfer doughnuts to paper towel-lined baking sheet to drain and cool slightly.