Pour 1/4 cup warm water into a small bowl. Stir in 2 1/4 teaspoons active dry yeast. Let sit for 5 minutes until foamy.

In a large bowl, combine warmed milk, 1/4 cup sugar, 2 large eggs, 1/2 teaspoon salt, and the foamy yeast. Whisk briefly.

Stir in 3 1/2 cups flour and the melted butter. Mix until a shaggy dough forms. If too sticky, add up to 1/2 cup more flour, a little at a time.

Turn dough onto a floured surface. Knead for 5–7 minutes until smooth and slightly elastic. The dough should be soft but not sticky.

Place dough in a greased bowl and cover with a clean towel or plastic wrap. Let it rise in a warm spot until doubled, about 1 hour.

Roll dough to about 1/2-inch thickness. Cut rings with a doughnut cutter or two round cutters (one large, one small). Re-roll scraps and cut more.

Transfer cut doughnuts to a parchment-lined tray, cover lightly, and let rise for 25–30 minutes until puffy.

Warm oil in a deep fryer or a heavy pot to 375°F. Use a thermometer for accuracy.

Fry doughnuts for 1–2 minutes per side until golden brown. Don’t overcrowd the pot; fry in batches.

Remove with a slotted spoon and drain on paper towels. While still warm, roll in granulated sugar or cool slightly and glaze/fill as desired.