Step 1

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. Scrape the seeds from the vanilla bean into the bowl. Whip for about 12 minutes at medium speed, or until the mixture has lightened and tripled in volume. Beat in the espresso extract. Transfer the mixture to a bowl placed in a larger bowl of ice water to maintain its consistency.

Step 2

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. Fold the whipped cream into the yolk mixture and return the bowl to the ice.

Step 3

Whip the egg whites in a mixer bowl or metal bowl until they are frothy. 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. Fold in the beaten egg whites into the egg yolk mixture until they are completely combined.

Step 4

Spoon the semifreddo into six 8- to 10-ounce coffee cups, leaving at least 1/2 inch at the top for the steamed milk. Gently tap the cups against the counter to level the mixture. Cover the cups with plastic wrap and place them in the freezer until frozen, at least 6 to 8 hours, or overnight. These will keep for up to 3 days in the freezer; after that time, they will start to deflate.

Step 5

For the cinnamon-sugar doughnuts part 1: Place the water in the bowl of a mixer. Crumble the yeast into the water; mix and crush the yeast with a spoon to dissolve. Add the flour to the bowl and, using the dough hook, mix slowly until the ingredients are thoroughly blended.

Step 6

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.

Step 7

Part 2: Crumble the yeast into the milk in a small bowl, stirring to dissolve. Place 3/4 cup of the flour, the sugar, and salt in the bowl of a mixer fitted with the dough hook. Mixing on low speed, pour in the milk and yeast mixture, followed by the egg yolks and butter. Mix for a minute to combine.

Step 8

Add the proofed sponge and the remaining 1/4 cup plus 2 tablespoons flour. Continue to beat at low speed until combined. 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. If the dough seems wet, it may be necessary to add 1 to 2 more tablespoons flour. Cover the bowl and let the dough proof overnight in the refrigerator.

Step 9

To shape the doughnuts: Place the chilled dough on a lightly floured surface and roll it out to a 1/2-inch thickness. 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). Place the doughnuts and doughnut holes on a baking sheet lined with parchment paper. Cover the doughnuts with a sheet of plastic wrap that has been sprayed with nonstick spray.

Step 10

At this point, the doughnuts can be frozen for several days, refrigerated, or allowed to rise at room temperature. If they're frozen, defrost them and then allow them to rise at room temperature for 1 to 1 1/2 hours. They can be refrigerated for several hours or overnight, to rise slowly. When you remove them from the refrigerator, uncover them and let them finish proofing in a warm place for 20 to 30 minutes. If they have not been refrigerated or frozen, they should rise at room temperature in about 15 minutes. Once proofed, they will have risen to approximately 3/4 inch.

Step 11

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. 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. Remove the doughnuts, drain them briefly on paper towels, and toss them in a bowl with the cinnamon sugar. Repeat with the remaining doughnuts.

Step 12

To complete: Remove the semifreddo from the freezer a few minutes before serving to soften slightly. Steam the milk using a cappuccino machine or frothing machine. Place a cup of semifreddo and a doughnut on each of six plates. Top each doughnut with a doughnut hole. Spoon the steamed milk over the semifreddo and serve immediately.