Step 1: In a large bowl or the bowl of a stand mixer, combine warm milk and 1 tablespoon of the sugar. Sprinkle yeast over the top and let sit for 5-10 minutes until foamy.
Step 2: Add the remaining sugar, sweet potato puree, melted butter, egg, vanilla, salt, and cinnamon. Whisk until smooth.
Step 3: Gradually add 3 cups of flour, mixing with a dough hook or wooden spoon. Add more flour 1/4 cup at a time until a soft dough forms and clears the sides of the bowl.
Step 4: Knead the dough for 5-7 minutes by hand or machine until smooth and elastic. Place in a greased bowl, cover, and let rise in a warm spot for 1.5 hours or until doubled.
Step 5: Punch down the dough and roll it out on a lightly floured surface to 1/2-inch thickness. Cut doughnuts using a 3-inch cutter.
Step 6: Place cut doughnuts on a parchment-lined baking sheet, cover, and let rise for another 30 minutes.
Step 7: Heat 2 inches of oil in a heavy pot to 350°F. Fry doughnuts for 60-90 seconds per side until golden brown. Drain on a wire rack.
Step 8: Whisk together powdered sugar, maple syrup, and cream to make the glaze. Dip warm doughnuts into the glaze and let set.