Combine flour, yeast, sugar, and salt in a stand mixer bowl. Add warm milk, egg yolks, and vanilla. Mix on low 2 minutes then medium for 4 minutes.
Add softened butter one tablespoon at a time with mixer running. Knead on medium 6 to 8 minutes until smooth and windowpane-ready.
Place dough in an oiled bowl, cover, and let rise at room temperature 1 to 1.5 hours until doubled.
Turn dough onto a floured surface and press to 1/2-inch thickness. Cut 3-inch rounds and 1-inch holes. Transfer to parchment-lined sheets. Proof loosely covered 30 to 45 minutes until puffed.
Heat oil to 355°F in a heavy pot. Fry 2 to 3 donuts at a time, 60 to 90 seconds per side until deep golden. Drain on a wire rack 2 minutes.
Whisk glaze ingredients until smooth. Dip warm donuts face-down, lift, let drip 3 seconds, and return to rack glaze-side up. Glaze sets within 5 minutes.