In a large saucepan over low heat, warm milk, sugar, and butter until butter has just begun to melt. Ensure mixture is warm, not hot, when mixed with sourdough starter to avoid destroying natural yeast.

In a large bowl combine milk mixture, sourdough starter, salt, and eggs. Using an electric mixer on medium speed, blend until ingredients are combined.

Add 2½ cups flour to the sourdough batter and then mix until a tacky dough forms.

Allow donut dough to rest for 10 minutes. Add a tablespoon or so of flour and knead for another 4-5 minutes.

Transfer donut dough to a lightly floured surface, kneading by hand until no longer sticky, forming a smooth elastic ball.

Place donut dough in greased bowl, covering with a damp kitchen towel or plastic wrap. Let dough rise in a warm place (room temperature ~ 70 degrees F) for 3-6 hours, or until doubled in size.

Punch out dough and transfer to a lightly floured surface, using a rolling pin. Roll dough out until approximately 1 inch thick. Usinig a donut cutter dipped in flour, cut out donuts and donut holes.

Place donut cut outs on a lightly floured surface, letting rest for 1 hour to become slightly puffy.

Preheat oil in a deep dutch oven to 350ºF. Gauge the oil temperature by using a candy thermometer.

While the oil is pre-heating, make the vanilla glaze. In a medium bowl, mix together the confectioner's sugar, whole milk, and vanilla extract until smooth. Set aside

Once oil has reached desired temperature, gently place doughnut cut-outs and doughnut holes in hot oil and fry for 1½ minutes per side then using a slotted spoon, remove doughnuts from the oil and place on paper towels to drain. Once drained, transfer to a wire rack over a sheet pan lined with parchment paper.

Cool donuts for 5 minutes, then dip donut in glaze, immersing it completely to coat. Place glazed donuts on cooling rack to dry.