Activate yeast by whisking 1 tsp sugar into warm milk, sprinkling yeast on top, and letting stand 5-10 minutes until foamy.
Add sourdough discard, eggs one at a time, melted butter, vanilla, remaining sugar, and salt. Mix until combined.
Add flour in two additions with nutmeg. Knead with dough hook for 6-8 minutes or by hand for 8-10 minutes until smooth, tacky, and passes the windowpane test.
Place dough in a greased bowl, cover, and let rise in a warm spot for 1-1.5 hours until doubled in size.
Turn dough onto a floured surface, deflate, and roll to 1/2 inch thickness. Cut out donuts using a donut cutter and place on parchment-lined baking sheets.
Cover the cut donuts and let them proof for 30-45 minutes until noticeably puffy and jiggly.
Heat oil in a Dutch oven to 360-375°F (182-190°C). Fry 2-3 donuts at a time for 1-1.5 minutes per side until deep golden brown. Drain on paper towels.
Whisk together powdered sugar, 4 tbsp milk, vanilla, and salt until smooth. Add remaining milk if needed for desired consistency.
Dip warm donuts into the glaze, let excess drip off, and set on a wire rack. Allow glaze to set for 5 minutes before serving.