In a small bowl, dissolve the yeast in warm water and let stand for 5 minutes until foamy. Meanwhile, combine the warmed milk, sugar, salt, egg, and melted butter in a large mixing bowl.

Add the yeast mixture to the milk mixture and stir to combine. Gradually add 3 cups of flour, stirring with a wooden spoon until a shaggy dough forms. Add remaining flour 1/4 cup at a time until the dough pulls away from the bowl sides but remains slightly tacky.

Turn the dough onto a lightly floured surface and knead for 5-7 minutes until smooth and elastic. Place in a greased bowl, cover with a clean kitchen towel, and let rise in a warm spot for 1 hour or until doubled in size.

Punch down the dough and roll it out on a floured surface to 1/2-inch thickness. Use a 3-inch donut cutter (or two round cutters) to cut out donuts and holes. Place them on a parchment-lined baking sheet, cover loosely, and let rest for 20 minutes.

Heat vegetable oil in a large, heavy pot or deep fryer to 350°F. Line a cooling rack with paper towels. Carefully slide 2-3 donuts into the hot oil without crowding, and fry for 1-2 minutes per side until golden brown. Use a slotted spoon to transfer to the prepared rack.

For the glaze, whisk together powdered sugar, milk, and vanilla extract in a medium bowl until completely smooth. The glaze should be thick but pourable—adjust consistency as needed.

Once donuts have cooled for 5 minutes, dip the top of each into the glaze, letting excess drip back into the bowl. Immediately top with rainbow sprinkles while the glaze is still wet. Place on a clean rack and let the glaze set for 10 minutes before serving.