In a large bowl or stand mixer, combine the warm milk and yeast. Let sit for 5 minutes until the mixture becomes frothy.

Add the flour, sugar, salt, egg, and softened butter to the yeast mixture. Mix until a sticky dough forms.

Knead the dough on a floured surface or with a dough hook for about 5-7 minutes until it becomes smooth and elastic.

Place the dough in a lightly greased bowl, cover with a towel, and let it rise in a warm place for about 1 hour, or until doubled in size.

Punch down the dough gently and roll it out on a floured surface to roughly 1/2-inch thickness.

Cut into donut shapes using a donut cutter or two round cookie cutters. Re-roll scraps once to cut more.

Place cut donuts on a parchment-lined baking sheet, cover loosely, and let rise again for 30-40 minutes until puffy.

Heat the vegetable oil in a deep pot or dutch oven to 350°F (175°C).

Fry the donuts in batches (avoid crowding) for about 1 minute per side until they are a light golden brown. Remove and drain on a wire rack.

While donuts cool slightly, whisk together powdered sugar, 1/4 cup milk, vanilla, and honey in a bowl until smooth.

Dip the warm donuts completely into the glaze, letting the excess drip off on a wire rack, and serve immediately.