Step 1

Heat milk in a small saucepan over medium just until warm to the touch (110°–115°). Transfer to a heatproof measuring glass. Top off with water to bring volume back to ¾ cup if needed. Immediately stir in yeast and a pinch of sugar; let sit until mixture begins to bubble and foam, about 4 minutes.

Step 2

Mix flour, salt, and remaining ¼ cup sugar in the bowl of a stand mixer fitted with the dough hook on low speed to combine. With motor running, pour in milk mixture, then add eggs and egg yolks one at a time, mixing well after each addition. Continue to mix, scraping down sides of bowl as needed, 3 minutes. (Dough should be workable but somewhat sticky at this point. You can add a little more flour by the tablespoon if needed, but resist the urge to add too much; you don’t want your dough to be tough.) Add 6 Tbsp. butter, a piece or 2 at a time, mixing after each addition until incorporated. (You might have to stop the mixer and knead butter in by hand to get it started.) Increase speed to medium and mix until dough is smooth, supple, and elastic (it will still be a little sticky), 6–8 minutes. (Alternatively, if you don’t have a stand mixer, mix dough with a wooden spoon in a large bowl, then knead in bowl about 6 minutes, dusting with additional flour if needed. Turn dough out onto a lightly floured surface and continue to knead by hand until smooth.)

Step 3

Lightly butter a large bowl and transfer dough to bowl. Cover with plastic wrap or a kitchen towel and let sit in a warm, draft-free spot until doubled in size, about 1 hour. (Alternatively, you can cover and chill at least 3 hours and up to 12 hours.)

Step 4

Meanwhile, bring milk and cardamom to a simmer in a medium saucepan over medium heat. Remove from heat, cover, and let sit to infuse at least 10 minutes and up to 1 hour.

Step 5

Whisk together egg yolks, cornstarch, sugar, and salt in a large bowl. Gradually whisk in a small amount of milk mixture to gently warm yolks. Pour egg mixture into remaining milk mixture in saucepan and bring to a boil over medium heat, stirring constantly and adjusting heat as needed to avoid scrambling eggs. Cook, stirring constantly and scraping bottom of pan, until mixture begins to thicken, 3–6 minutes, then whisk vigorously until thick and glossy, about 1 minute more. Strain custard through a fine-mesh sieve into a medium bowl. Cover with plastic wrap, pressing directly onto surface and chill until cold, about 2 hours.

Step 6

Turn out dough onto a lightly floured surface and roll out to ½" thick. Line 2 rimmed baking sheets with kitchen towels; dust towels well with flour. Punch out as many dough rounds as you can with cutter and transfer to prepared baking sheets. Gather scraps, reroll, and punch out additional rounds; transfer to baking sheets (you should have about 12 total). Lightly cover with plastic wrap and let sit in a warm, draft-free spot until nearly doubled in size and dough springs back slowly when gently pressed, at least 30 minutes and up to 2 hours, depending on temperature of your room (watch the dough, not the clock).

Step 7

When ready to fry, line 2 rimmed baking sheets (or a few plates) with paper towels. Fit thermometer to sides of a medium heavy pot and pour in oil to come 2" up sides. Heat over medium until thermometer registers 350°. Working in batches of 2 or 3 doughnuts at a time and adjusting heat as needed to maintain temperature of oil, carefully slide doughnuts into oil with a spider or slotted spoon and cook until golden brown, about 2 minutes per side. Transfer to prepared baking sheets and let cool slightly.

Step 8

Place sugar in a medium shallow bowl. Working one at a time, toss warm doughnuts in sugar and return to baking sheets. Let cool completely.

Step 9

Using a chopstick or the handle of a wooden spoon, poke a hole through side into each doughnut. (Be careful not to go all the way through.)

Step 10

Whisk heavy cream in a large bowl until soft peaks form. Uncover cardamom custard and whisk to loosen, then fold in whipped cream. Transfer filling to a pastry bag fitted with a small round tip (a resealable plastic bag with 1 corner cut off will also work) and gently pipe cardamom cream into doughnuts to fill.