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For the cappuccino semifreddo: Place the egg yolks and 1/4 cup plus 2 tablespoons of sugar in the bowl of a mixer fitted with the whisk attachment.
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Scrape the seeds from the vanilla bean into the bowl.
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Whip for about 12 minutes at medium speed, or until the mixture has lightened and tripled in volume.
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Beat in the espresso extract.
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Transfer the mixture to a bowl placed in a larger bowl of ice water to maintain its consistency.
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In a mixer bowl or a metal bowl, whip the heavy cream with 3 tablespoons of the sugar until it holds its shape when the whisk or beater is lifted.
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Fold the whipped cream into the yolk mixture and return the bowl to the ice.
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Whip the egg whites in a mixer bowl or metal bowl until they are frothy.
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While whipping, add in the remaining 3 tablespoons of sugar and whip the egg whites just until they hold soft peaks, being careful not to overwhip them.
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Fold in the beaten egg whites into the egg yolk mixture until they are completely combined.
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Spoon the semifreddo into six 8- to 10-ounce coffee cups, leaving at least 1/2 inch at the top for the steamed milk.
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Gently tap the cups against the counter to level the mixture.
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Cover the cups with plastic wrap and place them in the freezer until frozen, at least 6 to 8 hours, or overnight.
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These will keep for up to 3 days in the freezer; after that time, they will start to deflate.
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For the cinnamon-sugar doughnuts: Place the water in the bowl of a mixer.
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Crumble the yeast into the water; mix and crush the yeast with a spoon to dissolve.
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Add the flour to the bowl and, using the dough hook, mix slowly until the ingredients are thoroughly blended.
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Transfer this sponge to a bowl, cover, and let proof at room temperature for 1 to 2 hours, until it has doubled; or place the bowl in the refrigerator to proof overnight.
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Part 2: Crumble the yeast into the milk in a small bowl, stirring to dissolve.
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Place 3/4 cup of the flour, the sugar, and salt in the bowl of a mixer fitted with the dough hook.
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Mixing on low speed, pour in the milk and yeast mixture, followed by the egg yolks and butter.
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Mix for a minute to combine.
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Add the proofed sponge and the remaining 1/4 cup plus 2 tablespoons flour.
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Continue to beat at low speed until combined.
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Turn up the speed slightly and knead the dough for 4 to 5 minutes, or until it has formed a ball and cleans the sides of the bowl.
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If the dough seems wet, it may be necessary to add 1 to 2 more tablespoons flour.
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Cover the bowl and let the dough proof overnight in the refrigerator.
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To shape the doughnuts: Place the chilled dough on a lightly floured surface and roll it out to a 1/2-inch thickness.
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Cut out the doughnuts using a 2-inch doughnut cutter, or a 2-inch biscuit cutter (in which case, you will also need a 3/4-inch cutter to make the hole).
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Place the doughnuts and doughnut holes on a baking sheet lined with parchment paper.
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Cover the doughnuts with a sheet of plastic wrap that has been sprayed with nonstick spray.
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At this point, the doughnuts can be frozen for several days, refrigerated, or allowed to rise at room temperature.
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If they're frozen, defrost them and then allow them to rise at room temperature for 1 to 1 1/2 hours.
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They can be refrigerated for several hours or overnight, to rise slowly.
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When you remove them from the refrigerator, uncover them and let them finish proofing in a warm place for 20 to 30 minutes.
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If they have not been refrigerated or frozen, they should rise at room temperature in about 15 minutes.
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Once proofed, they will have risen to approximately 3/4 inch.
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To cook the doughnuts: In a deep heavy saucepan large enough to hold half the doughnuts and holes at a time, heat canola oil for deep-frying to 325 degrees F. Add half the doughnuts and holes to the oil and cook for approximately 30 seconds on the first side.
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Flip the doughnuts and fry for 1 minute on the second side, then turn back to the first side to cook for an additional 30 seconds, or until a deep golden brown.
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Remove the doughnuts, drain them briefly on paper towels, and toss them in a bowl with the cinnamon sugar.
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Repeat with the remaining doughnuts.
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To complete: Remove the semifreddo from the freezer a few minutes before serving to soften slightly.
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Steam the milk using a cappuccino machine or frothing machine.
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Place a cup of semifreddo and a doughnut on each of six plates.
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Top each doughnut with a doughnut hole.
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Spoon the steamed milk over the semifreddo and serve immediately.
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Note: Coffee extract can be made using equal amounts of espresso powder and boiling water.
