Sprinkle the yeast over the warm water; let stand 5 minutes, until foamy.
In a large bowl, whisk together sugar, salt, warmed milk, eggs, and melted butter. Stir in the yeast mixture.
Gradually add flour, mixing until a soft dough forms. Turn out onto a floured surface and knead 5–7 minutes until smooth and elastic.
Place the dough in a lightly oiled bowl, cover with plastic wrap, and let rise in a warm spot until doubled in size, about 1 hour.
Punch down the dough and roll to 1/2-inch thickness. Cut into rings using a donut cutter or two round cutters.
Transfer cut doughnuts to a parchment-lined tray, cover lightly, and let rest 30 minutes until puffed.
Heat 2 inches of oil in a heavy-bottomed pot to 375°F. Fry doughnuts in batches, 1 minute per side, turning once, until golden brown.
Drain on paper towels. Meanwhile, whisk glaze ingredients until smooth.
Dip warm doughnuts in glaze, let excess drip off, then place on a rack to set.