These are Italian-inspired filled doughnuts, not too far off from sufganiyot or jelly doughnuts, but daintier. Molly used to get them at a bomboloni shop she frequented in New York and loves to use pistachio pastry cream for the innards because of the lovely pale green color and delicate nutty flavor.
For the dough: In a large bowl or in the bowl of a stand mixer fitted with a dough hook, mix together the flour, sugar, yeast and salt. In a separate medium bowl, whisk together the water, oil and eggs. Pour the egg mixture into the flour mixture and mix on low to combine. Knead, either by hand on a floured surface or with a dough hook, for 7 to 10 minutes, adding more flour as necessary (it might be quite a bit, around 1 additional cup, but resist any urge to add too much!), until you have a smooth and still slightly sticky dough that just barely sticks to your hands if you give it a pinch.
Transfer the dough to a large, oiled bowl. Cover with plastic wrap and let it sit at room temperature until it has doubled in size, 1 1/2 to 2 hours. Alternatively, you can stick it in the refrigerator overnight and then let it sit at room temperature for about an hour before shaping.
For the pistachio pastry cream: While the dough rises, warm the milk in a small saucepan over low heat just until tiny bubbles form around the perimeter of the pan. In a medium bowl, whisk together the sugar, cornstarch and salt until smooth. Whisk in the egg yolks. While whisking the eggs, slowly drizzle in 1/3 cup of the warm milk to temper the yolks. Pour the entire egg mixture into the saucepan of warm milk and cook over low heat, stirring constantly, until a few bubbles pop to the surface and the mixture is thick like pudding, 2 to 3 minutes. Add the pistachios, vanilla and almond extracts and food coloring, if using, and stir to combine. Transfer the cream to a clean bowl. Cover the surface with plastic wrap and refrigerate until cold, at least 2 hours. (It can also be made a day ahead.)
When the dough has doubled, line a baking sheet with parchment paper. Punch the dough down and roll out on a floured surface to about 1/2-inch thickness. Use a 2-inch cutter to cut as many rounds as possible; you should get about 18 rounds. Place the rounds on the baking sheet. (You can re-roll the scraps once more, if necessary, to get 18, but re-rolling more than once will make the bomboloni tough.) Cover loosely with plastic wrap and let rise until puffy, up to 30 minutes (or a little less if your kitchen is very warm).
Meanwhile, heat about 2 inches of neutral oil in a Dutch oven to 350 degrees F. Set a cooling rack over a rimmed baking sheet. Spread some sugar in a small bowl. Fry the rounds in 2 batches, turning once, until puffed and golden brown on both sides, 3 to 4 minutes total per batch. Remove to the cooling rack. While still hot, roll in the sugar to fully coat. Let cool completely.
Put the chilled pastry cream in a pastry bag fitted with a small plain tip. To fill, insert the tip straight down in the top of a bombolone and wiggle the tip to make a little space. Fill the space with the pastry cream. Fill all of the bomboloni, then garnish with pistachios and serve right away.
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