1. To make the doughnuts: Fill a large saucepan fitted with a steamer basket or rack with water to a depth of 3 inches. Bring the water to a gentle simmer and add the sweet potato to the steamer basket. Cover and steam until a knife pierces through the flesh easily, about 20 minutes. Remove from the heat and cool slightly.
2. Mash the sweet potato with a fork until smooth. Set aside.
3. Sprinkle the yeast over ¼ cup warm water (110°F) and let sit until foamy, about 10 minutes.
4. Transfer the mashed sweet potato to the bowl of an electric mixer fitted with the paddle attachment and add the evaporated milk, buttermilk, and 1/3 cup water. Mix on medium speed until well incorporated, about 3 minutes. Add the sugar and mix until dissolved, about 5 minutes. Turn the mixer speed to low and add the foamy yeast, then add the flour ½ cup at a time. Once all the flour has been incorporated, mix for 1 minute more. Cover the bowl with plastic wrap and let the dough rise in a warm place until nearly doubled in bulk, about 1 hour.
5. Meanwhile, make the filling: Toss the apples with the lemon juice, sugar, and cinnamon. Set a large skillet over medium heat for 3 minutes, then add the apples and butter. Cook, stirring occasionally, until the apples are browned and slightly softened, about 30 minutes. Stir in the vanilla extract and set aside to cool completely.
6. Generously butter twenty-four 4-ounce ramekins or two 12-cup muffin tins, then sprinkle the molds lightly with sugar. Fill each mold with 3 tablespoons of the soft, wet dough, then press a piece of roasted apple into the center of each. Gently gather the dough together to cover the apple and pinch the top seam tightly. Cover lightly with plastic wrap and let rise for 30 minutes.
7. Meanwhile, preheat the oven to 350°F.
8. Uncover the doughnuts and bake until puffed and golden brown, about 20 minutes. Serve warm.
Kabocha Squash Beignets: Omit the butter and sugar for the molds. In the doughnut recipe, omit the Roast Apple Filling and substitute 1 1/3 cups (12¼ ounces/343 grams) mashed Kabocha squash (see page 18) for the sweet potato and pumpkin spiced ale for the buttermilk. You will need canola, vegetable, or other neutral oil for deep-frying. Fill a saucepan with oil to a depth of 3 inches and heat to 350°F. When the oil is ready (a tiny pinch of flour will sizzle), shape the risen dough into 2-inch balls and gently drop into the oil; cook about 5 at a time—do not crowd the pan. Cook until puffy and golden brown, about 2 minutes, then turn and cook for another minute. Remove the cooked beignets with a slotted spoon and drain on paper towels. Roll the hot beignets in sugar, if desired, and serve warm.