Activate Yeast: Combine warm milk and yeast in a bowl. Let sit for 5-10 minutes until foamy.
Mix Dough: In a stand mixer, combine flour, sugar, and salt. Add the yeast mixture, eggs, and vanilla. Knead until smooth. Incorporate butter slowly, kneading for another 5-7 minutes.
Proof: Cover and let rise in a warm spot for 1-1.5 hours until doubled.
Shape: Roll dough to 1/2-inch thickness. Cut out circles using a donut cutter. Let them rest for 30 minutes on a floured surface.
Fry: Heat oil to 365°F (185°C). Fry donuts for 60-90 seconds per side until golden brown.
Glaze: Whisk all glaze ingredients together. Dip warm (not hot) donuts into the glaze and set on a wire rack to set.