Activate yeast: In a large bowl, combine warm milk, 2 tbsp sugar, and yeast. Let sit for 5–10 minutes until foamy.
Mix wet ingredients: Add eggs, remaining ¼ cup sugar, vanilla, and butter to the activated yeast. Mix until smooth.
Add dry ingredients: Gradually add flour and salt. Mix until dough forms.
Knead: Knead on a floured surface for about 8–10 minutes (or use a stand mixer with dough hook for 5 minutes) until smooth and slightly tacky.
First rise: Place dough in a lightly oiled bowl, cover, and let rise in a warm place until doubled—about 1 to 1½ hours.
Roll dough to about ½ inch thick. Cut into 2 ½–3 inch rounds. Place on parchment-lined tray. Cover and let rise again until puffy, about 45 minutes.
Fry: Heat oil to 340–350°F (170–175°C). Fry 2–3 donuts at a time, flipping once, until deep golden brown. Drain on paper towels.
Sugar coat: While warm, roll in granulated sugar.