Step 1: In a large bowl, combine warm milk, 1 tablespoon of sugar, and the yeast. Let it sit for 5-10 minutes until it becomes foamy and active.
Step 2: Stir in the remaining sugar, melted butter, egg, vanilla, and salt. Gradually add the flour, mixing until a dough forms.
Step 3: Knead the dough on a floured surface for 8-10 minutes until smooth and elastic. Place in a greased bowl, cover, and let rise in a warm place for 1 hour or until doubled in size.
Step 4: Punch down the dough and roll it into a large rectangle (approx. 12×18 inches). Spread softened butter over the surface, then sprinkle with the brown sugar and cinnamon.
Step 5: Roll the dough tightly into a log and slice into 12 even rounds. Place on a parchment-lined tray, cover, and let rise for another 30-45 minutes.
Step 6: Heat oil in a heavy-bottomed pot to 350°F. Carefully fry each donut for 1-2 minutes per side until golden brown. Drain on a wire rack.
Step 7: Whisk powdered sugar and 2 tablespoons of milk together to create a glaze. Dip the warm cinnamon roll donuts into the glaze and serve immediately.