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For Florentine Doughnuts:
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Place the warm water in a small bowl and sprinkle the yeast over it.
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Add 1/2 teaspoon granulated sugar and stir to dissolve the yeast.
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Set the bowl aside and allow the yeast to proof until foamy, about 4 minutes.
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5
Place the milk in a small saucepan or a microwaveable container and heat until warm to the touch.
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Place the all purpose flour, cake flour, remaining 1 1/4 cups granulated sugar, salt and spices in the bowl of an electric mixer and use the paddle attachment to combine them thoroughly.
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Make a well in the center of the bowl and add the yeast mixture, warm milk, and eggs.
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Beat on medium until a smooth dough forms.
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Beat in the butter, then remove the paddle and beat in enough additional flour on low speed to make a smooth, soft, somewhat elastic dough.
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Switch to medium speed and mix for 2 to 3 minutes.
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Turn the dough into a greased bowl and cover with plastic wrap.
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Place the bowl in a warm, draft-free spot and allow the dough to rise until doubled, about 2 hours.
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13
Gently deflate the dough by pulling it from the sides of the bowl and turn it onto a floured surface.
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Pat or roll the dough to a thickness of about 1/2 inch.
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15
Using a fluted or plain-edged round cutter 1 to 1 1/2 inches in diameter, cut the doughnuts.
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Place them on a baking sheet dusted with flour and cover them with plastic wrap.
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Allow them to proof while you heat the oil.
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Heat 6 inches (4 to 6 cups) olive oil to 360F in a large, heavy-bottomed pot.
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Fry the doughnuts in batches, 4 or 5 at a time, in the hot oil, gently turning them until they are golden brown on both sides and cooked in the center.
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Drain the doughnuts on paper towels.
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While they are still warm, sprinkle them generously with confectioners sugar on both sides.
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They are best served warm, with vanilla custard alongside for dipping.
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For Vanilla Custard:
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Pour the milk and cream into a medium saucepan.
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Split the vanilla bean lengthwise, scrape the seeds with the blunt side of a small knife, and add them to the pan along with the bean and vanilla extract.
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Add 1/2 cup sugar and place the pan over medium heat.
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Meanwhile, place the egg yolks in a large bowl.
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Whisk in the remaining 1/4 cup and 1 tablespoon sugar, followed by the salt and cornstarch.
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When the milk mixture comes to a boil, remove the pan from the heat.
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Slowly pour some of the mixture into the egg yolks, whisking constantly.
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Add the rest of the hot liquid to the eggs, whisking constantly, then transfer the mixture back to the saucepan and place it over low heat.
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Keep whisking it to prevent the custard from scorching or overcooking.
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When the custard has thickened somewhat, remove it from the heat and strain it through a chinois or fine-meshed sieve into a bowl.
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Place the bowl in a larger bowl of ice water to chill it, whisking occasionally as the custard cools.
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Press a piece of plastic wrap onto the surface of the custard to prevent a skin from forming.
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Refrigerate the custard in an airtight container until ready to serve.
