Mix lukewarm milk, yeast, sugar, egg, and melted cooled butter. Add flour and salt, then knead 8–10 minutes until smooth and slightly tacky.
Place in a lightly greased bowl, cover, and rise 60–90 minutes until roughly doubled. Prepare a tray with small parchment squares for shaped donuts.
Roll dough to 1/2-inch thickness on a lightly floured surface. Cut twelve 3-inch rings with 1-inch centers, gently rerolling scraps once if needed.
Transfer rings and holes to the tray, cover loosely, and rise 30–45 minutes until puffy. Avoid handling repeatedly during this second rise.
Heat 2 inches of oil in a heavy pot to 350°F (175°C), leaving ample empty space above it. Use a thermometer and keep the pot stable.
Fry two or three donuts at a time for 60–90 seconds per side, until golden. Slide off the paper before frying; maintain oil temperature and fry holes more briefly.
Lift with a slotted utensil onto a rack over a sheet, then cool 5 minutes. Do not add wet utensils or water to hot oil.
Whisk glaze ingredients until smooth and dip the warm donuts. Let excess drip onto the sheet and allow glaze to settle before serving. Cool the oil completely before handling it.
