Mix warm milk, eggs, sugar, vanilla, and melted butter in a stand mixer
Add yeast, flour, and salt; knead 7–9 minutes until sticky but pulling from the bowl
Cover and let rise until doubled (1–3 hours)
Roll dough to ½" thick; cut 3" rounds and 1" centers
Place each doughnut on a parchment square and let rise 30–60 minutes until puffed
Heat oil to 350°F; fry doughnuts 2–3 minutes per side until deep golden
Transfer to paper towels to drain
Whisk glaze ingredients until smooth
Cool doughnuts 5 minutes, dip in glaze, and place on a rack to set
