1. Make the doughnuts: Heat butter in a medium saucepan over medium heat until dark brown and fragrant, about 10 minutes. Remove from heat, and stir in milk. Let cool slightly.

2. Combine yeast, warm water, and 1/4 teaspoon granulated sugar in a mixer bowl. Let stand until foamy, about 5 minutes.

3. Whisk sweet-potato puree, egg, yolk, vanilla, salt, cinnamon, and remaining 3 tablespoons granulated sugar in a medium bowl. Whisk in butter mixture, then add to yeast mixture in mixer bowl. Attach bowl to mixer fitted with the dough-hook attachment, and mix on low speed. Slowly add flour until incorporated. Raise speed to medium-high, and beat until dough becomes elastic, about 3 minutes. Dough will be very soft.

4. Transfer dough to an oiled bowl, and cover with plastic wrap. Let stand until double in size, about 2 hours. Punch dough, then cover and refrigerate for at least 2 hours.

5. On a generously floured surface, roll out dough to a 1/4-inch thickness. Using a 1 1/2-inch round cutter, cut out rounds. Using the wide end of a 1/2-inch round pastry tip, cut holes in center of each round. Transfer doughnuts and holes to generously floured baking sheets, and cover loosely with plastic. Let rise in a warm place for 45 minutes.

6. Line wire racks with several layers of paper towels. Heat 4 inches of oil in a large, heavy pot until it reaches 365 to 370 degrees on a deep-fry thermometer. Fry doughnuts and holes in small batches until golden brown, about 20 seconds per side. (Adjust heat as needed to maintain temperature of oil.) Using a slotted spoon or a wire-mesh skimmer, transfer doughnuts and holes to paper towels to drain. Let cool slightly.

7. Make the glaze: Heat butter in a medium saucepan until dark brown, about 10 minutes. Whisk in confectioners' sugar until a smooth paste forms, then gradually whisk in milk until thick and silky. Let stand for 10 minutes before using, whisking occasionally to keep glaze smooth.

8. Pour glaze into a large, deep bowl. With tongs, dip doughnuts and holes into glaze, turning to coat. Remove, letting excess drip off before transferring to clean wire racks set over baking sheets. Let glaze harden before serving, about 15 minutes. Serve immediately.